Integrals: Definition, Evaluation, and Applications in Biology

Question 1

Which of the following is **not** a valid application of integrals in biology?

Accepted Answer

Predicting future events

Answer

Estimating population growth

Answer

Calculating the area under a curve

Answer

Solving differential equations

Question 2

Does the improper integral ∫[0,∞] e^(-x^2) dx converge or diverge?

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Converges conditionally

Question 3

Which of the following statements accurately describes a fundamental theorem of calculus?

Accepted Answer

The integral of a derivative is the original function.

Answer

The integral of a derivative is always zero.

Answer

The derivative of an indefinite integral is equal to the original integrand.

Answer

The area under a curve can be calculated using integration.

Question 4

Which integral represents the area under the curve y = sin(x) from x = 0 to x = π?

Accepted Answer

∫[0,π] sin(x) dx

Answer

∫[0,π] cos(x) dx

Answer

∫[π,0] sin(x) dx

Answer

∫[0,2π] sin(x) dx

Question 5

Which of the following methods is used to evaluate improper integrals?

Accepted Answer

Convergence test

Answer

Substitution method

Answer

Partial fractions

Answer

Integration by parts

Question 6

Which integral calculates the work done by a force over a given distance?

Accepted Answer

∫[a,b] f(x) dx

Answer

∫[0,1] f(x) dx

Answer

∫[a,∞] f(x) dx

Answer

∫[0,∞] f(x) dx

Question 7

Which differential equation models exponential population growth?

Accepted Answer

dy/dt = ky

Answer

d^2y/dt^2 = ky

Answer

dy/dt = ky^2

Answer

d^2y/dt^2 = ky^2

Question 8

Evaluate the integral ∫(e^x + sin(x)) dx:

Accepted Answer

e^x - cos(x) + C

Answer

e^x + cos(x) + C

Answer

(e^x + sin(x))x + C

Answer

e^xsin(x) + C

Question 9

How have integrals revolutionized our understanding of biological processes? Provide an example of how mathematical modeling using integrals has deepened our comprehension of living systems.

Accepted Answer

Integrals enable us to calculate population changes over time, advancing population ecology.

Answer

Integrals have no practical applications in biology and are only used for theoretical calculations.

Answer

Integrals are limited to calculating areas and volumes of biological structures.

Answer

Integrals have exclusively contributed to genetics and have no relevance to other biological disciplines.

Question 10

Which of the following substitutions is most advantageous for integrating the function f(x) = (x^2 + 1)/(x + 1)?

Accepted Answer

u = x + 1

Answer

u = x^2 + 1

Answer

u = x^2

Answer

u = 1/(x + 1)

Question 11

Evaluate the improper integral: ∫₀^∞ e^(-x) dx

Accepted Answer

1

Answer

∞

Answer

0

Answer

-1

Question 12

Find the center of mass of a thin rod of length L and linear density ρ(x) = kx, where k is a constant.

Accepted Answer

L/3

Answer

L/4

Answer

L/2

Question 13

Which of the following is the indefinite integral of (x^2 + 3x - 2)?

Accepted Answer

((x^3)/3) + ((3x^2)/2) - 2x + C

Answer

((x^3)/3 + (3x^2) - 2x + C

Answer

(x^3 + 3x^2 - 2) + C

Answer

((x^2 + 3x) / 2) + C

Question 14

What is the definite integral of (sin x) from 0 to π/2?

Accepted Answer

1

Answer

-1

Answer

0

Answer

π/2

Question 15

Which of the following integration techniques involves substituting a trigonometric function for a more complex expression under the square root?

Accepted Answer

Trigonometric Substitution

Answer

Partial Fractions

Answer

U-Substitution

Answer

Integration by Parts

Question 16

What is the integral of (e^x) + (x^2) dx?

Accepted Answer

(e^x) + ((x^3)/3) + C

Answer

((e^x) * x^2) + C

Answer

(e^x) - ((x^3)/3) + C

Answer

(e^x) * (x^2) + C

Question 17

Which integration technique involves breaking down a rational fraction into a sum of simpler fractions?

Accepted Answer

Partial Fractions

Answer

U-Substitution

Answer

Integration by Parts

Answer

Trigonometric Substitution

Question 18

Find the value of the definite integral of the function f(x) = sin(x) from x = 0 to x = π/2.

Accepted Answer

1

Answer

0

Answer

-1

Answer

π/2

Question 19

Calculate the indefinite integral of the function f(x) = e^x + ln(x).

Accepted Answer

e^x + xln(x) - x + C

Answer

e^x - xln(x) + x + C

Answer

e^x + xln(x) + x + C

Answer

e^x - xln(x) - x + C

Question 20

Which integral represents the area under the curve y = x^2 from x = 0 to x = 2?

Accepted Answer

∫[0,2] x^2 dx

Answer

∫[2,0] 2x dx

Answer

∫[2,0] x^2 dx

Answer

∫[0,2] 2x dx

Question 21

Which of the following mathematical expressions represents the area under the curve of a function f(x) over the interval [a, b]?

Accepted Answer

∫[a,b] f(x) dx

Answer

f(x) / (b - a)

Answer

f(x) * (b - a)

Answer

∑[a,b] f(x) Δx

Question 22

Which mathematical technique is primarily used to evaluate definite integrals?

Accepted Answer

Fundamental Theorem of Calculus

Answer

Substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Question 23

Identify the improper integral that converges.

Accepted Answer

∫[1, ∞] (1/x^2) dx

Answer

∫[0, π/2] (tan x) dx

Answer

∫[-1, 1] (1/(x-1)) dx

Answer

∫[0, 1] (1/x) dx

Question 24

The integral ∫[0,∞] e^(-x²) dx is known as which special type of integral?

Accepted Answer

Gaussian integral

Answer

Cauchy integral

Answer

Fresnel integral

Answer

Laplace integral

Question 25

What is the indefinite integral of the polynomial function  f(x) = x² + 2x - 3?

Accepted Answer

x³/3 + x² - 3x + C

Answer

x³ - 2x² - 3x + C

Answer

x³ + 2x² - 3x + C

Answer

x² + 2x - 3 + C

Question 26

Using integration by parts, determine the indefinite integral of e^(2x) + cos(x).

Accepted Answer

(1/2)e^(2x) + sin(x) + C

Answer

2e^(2x) - sin(x) + C

Answer

2e^(2x) + sin(x) + C

Answer

(1/2)e^(2x) - sin(x) + C

Question 27

What is the indefinite integral of 1/(x² + 1) with respect to x?

Accepted Answer

arctan(x) + C

Answer

tan(x) + C

Answer

1/x + C

Answer

x + C

Question 28

Calculate the area bounded by the curve y = x³ - 2x² + 1 and the x-axis between x = 0 and x = 2.

Accepted Answer

4/3

Answer

8/3

Answer

16/3

Answer

2/3

Question 29

Determine the volume of the solid formed by revolving the region enclosed by the curves y = x² and y = 4 around the x-axis.

Accepted Answer

(32π/5) cubic units

Answer

(64π/5) cubic units

Answer

(16π/5) cubic units

Question 30

What is a significant advantage of using numerical integration methods?

Accepted Answer

Approximating integrals for complex functions that lack a closed-form antiderivative.

Answer

Always providing exact solutions for integrals.

Answer

Being computationally faster than analytical integration methods.

Question 31

Calculate the center of mass of a uniform rod with length L and density ρ.

Accepted Answer

L/2

Answer

3L/4

Answer

L/4

Question 32

Which of the following scenarios demonstrates a real-world application of integrals in biology?

Accepted Answer

Determining the population growth rate based on a given differential equation.

Answer

Predicting the trajectory of a launched projectile.

Answer

Finding the average weight of a fish within a population.

Question 33

What is the indefinite integral of the function f(x) = 3x² + 2x - 1?

Accepted Answer

x³ + x² - x + C

Answer

9x + 2 + C

Answer

x³ + x² - x

Answer

6x + 2 + C

Question 34

The area enclosed by the curve of a function f(x) between x = a and x = b is represented by which definite integral?

Accepted Answer

∫[a,b] f(x) dx

Answer

f'(x)

Answer

∫[b,a] f(x) dx

Answer

d/dx [∫[a,b] f(x) dx]

Question 35

The rate of change of the population of a bacterial colony is given by the function dP/dt = 2t + 1, where P is the population and t is time in hours. Find the population after 3 hours if the initial population was 100.

Accepted Answer

118

Answer

127

Answer

103

Answer

109

Question 36

What is the integral of 1/(x+1) dx?

Accepted Answer

ln|x+1| + C

Answer

(x+1)² + C

Answer

ln(x+1) + C

Answer

1/(x+1)² + C

Question 37

The volume of a solid generated by rotating the curve y = f(x) from x = a to x = b around the x-axis is given by which integral?

Accepted Answer

π∫[a,b] (f(x))² dx

Answer

∫[a,b] f'(x) dx

Answer

∫[a,b] (f(x))² dx

Answer

π∫[a,b] f(x) dx

Question 38

What is the average value of the function f(x) = x² from x = 0 to x = 2?

Accepted Answer

4/3

Answer

8/3

Answer

1

Answer

2/3

Question 39

Which of the following is NOT a property of definite integrals?

Accepted Answer

∫[a,b] f(x) dx = ∫[b,a] f(x) dx

Answer

∫[a,c] f(x) dx + ∫[c,b] f(x) dx = ∫[a,b] f(x) dx

Answer

∫[a,b] (f(x) + g(x)) dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx

Answer

∫[a,b] k*f(x) dx = k∫[a,b] f(x) dx

Question 40

The integral of a rate of change function represents:

Accepted Answer

The total accumulation over a given interval.

Answer

The instantaneous rate of change at a specific point.

Answer

The average rate of change over a given interval.

Answer

The maximum value of the rate of change function.

Question 41

Which of the following is NOT a technique for evaluating indefinite integrals?

Accepted Answer

L'Hopital's Rule

Answer

Substitution

Answer

Integration by Parts

Answer

Partial Fractions

Question 42

The area under the curve of a function f(x) from x = a to x = b is represented by:

Accepted Answer

The definite integral of f(x) from a to b.

Answer

The derivative of f(x) evaluated at x = b.

Answer

The indefinite integral of f(x) from a to b.

Answer

The average value of f(x) between a and b.

Question 43

Which of the following is the correct antiderivative of the function f(x) = cos(x)?

Accepted Answer

sin(x) + C

Answer

cos(x) + C

Answer

-sin(x) + C

Answer

-cos(x) + C

Question 44

The definite integral of the function f(x) = 2x from x = 1 to x = 3 represents:

Accepted Answer

The area of the region bounded by the curve of f(x), the x-axis, and the lines x = 1 and x = 3.

Answer

The slope of the tangent line to the curve at x = 2.

Answer

The volume of the solid generated by rotating the region about the x-axis.

Question 45

The integral of a velocity function with respect to time represents:

Accepted Answer

The displacement of an object.

Answer

The acceleration of an object.

Answer

The instantaneous velocity of an object.

Answer

The average velocity of an object.

Question 46

Which of the following represents the indefinite integral of f(x) = 2x^3 - 5x + 1?

Accepted Answer

(x^4)/2 - (5x^2)/2 + x + C

Answer

(x^4)/2 + 5x^2 + x + C

Answer

2x^3 - 5x + 1 + C

Answer

(2x^4)/3 - (5x^2)/2 + x + C

Question 47

Which of the following is the integral of f(x) = e^(2x)?

Accepted Answer

(e^(2x))/2 + C

Answer

2e^(2x) + C

Answer

e^(4x) + C

Answer

e^(2x) + C

Question 48

Find the integral of f(x) = sin(x)cos(x).

Accepted Answer

(sin^2(x))/2 + C

Answer

sin(2x) + C

Answer

(cos^2(x))/2 + C

Answer

cos(2x) + C

Question 49

Which of the following is the integral of f(x) = 1/(x^2 + 1)?

Accepted Answer

arctan(x) + C

Answer

1/(x^2 + 1) + C

Answer

-arctan(x) + C

Answer

x/(x^2 + 1) + C

Question 50

Calculate the integral of f(x) = x * sqrt(x^2 + 1).

Accepted Answer

(1/3)*(x^2 + 1)^(3/2) + C

Answer

x*(x^2 + 1) + C

Answer

(1/2)*(x^2 + 1)^(1/2) + C

Answer

(3/2)*(x^2 + 1)^(2/3) + C

Question 51

Find the integral of f(x) = (x + 1)/(x^2 + 2x + 3).

Accepted Answer

ln(|x^2 + 2x + 3|) + C

Answer

(x + 1) * ln(|x^2 + 2x + 3|) + C

Answer

-ln(|x^2 + 2x + 3|) + C

Answer

1/(x^2 + 2x + 3) + C