Advanced Integration Techniques And Geometric Applications Practice Tests

Question 1

Evaluate the indefinite integral $$\int 8x(4x^2-3)^3 dx$$ using the substitution $u = 4x^2-3$.

Accepted Answer

$\frac{1}{4}(4x^2-3)^4 + C$

Answer

$(4x^2-3)^4 + C$

Answer

$\frac{1}{2}(4x^2-3)^4 + C$

Answer

$\frac{1}{8}(4x^2-3)^4 + C$

Question 2

When evaluating the definite integral $$\int_0^1 x(x^2+1)^2 dx$$ using the substitution $u = x^2+1$, what are the new limits of integration for $u$?

Accepted Answer

$u=1$ to $u=2$

Question 3

Which of the following is the correct transformation of $$\int_0^{\pi/2} \sin^3(x) \cos(x) dx$$ using $u = \sin(x)$?

Accepted Answer

$\int_0^1 u^3 du$

Answer

$\int_0^{\pi/2} u^3 du$

Answer

$\int_0^1 u^4 du$

Answer

$\int_0^1 \frac{u^4}{4} du$

Question 4

To evaluate $$\int x \sqrt{x+1} dx$$, one can use the substitution $u = x+1$, which implies $x = u-1$ and $dx = du$.

Accepted Answer

True

Answer

False

Question 5

Calculate the value of $$\int_0^1 2x(x^2+1)^3 dx$$ using substitution.

Accepted Answer

$\frac{15}{4}$

Answer

$\frac{17}{4}$

Answer

4

Answer

$\frac{7}{2}$

Question 6

Which of the following integrals require a constant adjustment (multiplying/dividing by a constant) when performing a standard $u$-substitution?

Answer

$\int x^2(x^3+5)^4 dx$

Answer

$\int (x+1)^2 dx$

Answer

$\int \sin(2x) dx$

Question 7

Evaluate $$\int \frac{\cos(\ln x)}{x} dx$$ using the substitution $u = \ln x$.

Accepted Answer

$\sin(\ln x) + C$

Question 8

Evaluate the indefinite integral $\int 	an(x) \sec^2(x) \, dx$ using the substitution $u = 	an(x)$.

Accepted Answer

$\frac{1}{2} 	an^2(x) + C$

Question 9

When evaluating $\int_0^{\pi/4} \sin^3(x) \cos(x) \, dx$ using $u = \sin(x)$, what are the new limits of integration?

Accepted Answer

Option A

Answer

Option B

Answer

Option C

Answer

Option D

Question 10

Solve $\int \frac{x}{\sqrt{x^2+1}} \, dx$ using the substitution $u = x^2+1$.

Accepted Answer

$\sqrt{x^2+1} + C$

Question 11

Which of the following substitutions are appropriate for evaluating $\int \frac{\cos(x)}{\sin^2(x)} \, dx$?

Accepted Answer

Option A

Answer

Option B

Answer

Option C

Answer

Option D

Question 12

True or False: To evaluate $\int_1^e \frac{(\ln x)^2}{x} \, dx$ with $u = \ln(x)$, the new limits of integration are $u=1$ and $u=e$.

Accepted Answer

False

Answer

True

Question 13

Find the value of $\int_0^{\pi/2} \cos(x) e^{\sin(x)} \, dx$.

Accepted Answer

$e - 1$

Question 14

Find the area of the region bounded by the curves $y = x^2$ and $y = 4x - x^2$.

Accepted Answer

8/3

Answer

4/3

Answer

2

Answer

4

Question 15

Set up, but do not evaluate, the integral with respect to $y$ to find the area of the region bounded by $x = y^2$ and $x = y + 2$.

Accepted Answer

\int_{-1}^2 ((y + 2) - y^2) dy

Question 16

A solid is formed by revolving the region bounded by $y = \sqrt{x}$, $y = 0$, and $x = 4$ about the x-axis. Using the disk method, which integral represents the volume?

Accepted Answer

$\pi \int_0^4 x dx$

Answer

$\pi \int_0^4 \sqrt{x} dx$

Answer

$\pi \int_0^2 x^2 dx$

Answer

$\pi \int_0^4 x^2 dx$

Question 17

When using the washer method to find the volume of a solid revolved about the x-axis, the cross-sectional area is given by $A(x) = \pi(R(x)^2 - r(x)^2)$, where $R(x)$ is the outer radius and $r(x)$ is the inner radius.

Accepted Answer

True

Answer

False

Question 18

Using the method of cylindrical shells, what is the volume of the solid formed by revolving the region bounded by $y = x - x^2$ and $y = 0$ about the y-axis?

Accepted Answer

$\pi/2$

Answer

$\pi/6$

Answer

$\pi/4$

Answer

$\pi/3$

Question 19

Which of the following conditions are necessary to use the method of cylindrical shells for a region bounded by $f(x)$ and $g(x)$ revolved about the y-axis?

Answer

The radius of the shell is $x$

Answer

The height of the shell is $|f(x) - g(x)|$

Answer

The integration is performed with respect to $x$

Answer

The integration is performed with respect to $y$

Question 20

Determine the volume of the solid generated by revolving the region bounded by $y = x^2$ and $y = 1$ about the line $y = 2$ using the washer method.

Accepted Answer

56\pi/15

Advanced Integration Techniques And Geometric Applications Practice Tests

Practice Tests

Advanced Integration Techniques and Geometric Applications

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