Unit 01 - Essential problems

Shells

01

Shells volume - offset graph, $y$-axis

Consider the region in the first quadrant bounded by the lines $x=0$, $x=2$, $y=0$, and the curve $y=\frac{1}{\sqrt{x^2+1}}$. Revolve this about the $y$-axis.

Find the volume of the resulting solid.

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Solution

Solutions---0030-01

IBP

02

Integration by parts - A and T

Compute the integral:

$$\int x^2\sin xdx$$

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Solution

Solutions---0010-02

03

Integration by parts - A and L

Compute the integral:

$$\int x^3\ln xdx$$

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Solution

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05

Integration by parts - A and I

Compute the integral:

$$\int {\tan }^{-1}(x)dx$$

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Solution

Solutions---0010-05

Trig power products

01

Somewhat odd power product

Compute the integral:

$$\int {\sin }^{2}x\cdot {\cos }^{3}xdx$$

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Solution

Solutions---0040-01

02

Tangent and secant both even

Compute the integral:

$$\int {\tan }^{2}x\cdot {\sec }^{2}xdx$$

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Solution

Solutions---0040-02

05

Tangent and secant mixed parity

Compute the integral:

$$\int {\tan }^{3}x{\sec }^{2}xdx$$

(a) Using $du={\sec }^{2}xdx$.

(b) Using $du=\sec x\tan xdx$.

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Solution

Solutions---0040-05

Trig subs

01

Trig sub

Compute the definite integral:

$${\int }_0^{1/2}\frac{x^2}{\sqrt{1-x^2}}dx$$

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Solution

Solutions---0050-01

03

Trig sub

Compute the integral:

$$\int \frac{dx}{x^3\sqrt{x^2-4}}$$

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Solution

Solutions---0050-03

05

Trig sub

Compute the integral:

$$\int \frac{x^2}{{(x^2+1)}^{3/2}}dx$$

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Solution

Solutions---0050-05

Partial fractions

01

Distinct linear factors

Compute the integral:

$$\int \frac{1}{(x+2)(x-3)}dx$$

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Solution

Solutions---0060-01

02

Long division first

Compute the integral:

$$\int \frac{2x^3+3x^2+7x+4}{x+1}dx$$

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Solution

Solutions---0060-02

07

Partial fractions - linear and quadratic

Compute the integral:

$$\int \frac{5x^2-5x+14}{(x-2)(x^2+4)}dx$$

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Solution

Solutions---0060-07

08

Partial fractions - repeated factor

Compute the integral:

$$\int \frac{1}{x{(x-1)}^3}dx$$

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Solution

Solutions---0060-08

Simpson’s Rule

02

Simpson’s Rule for volume by shells

Use Simpson’s Rule with $n=6$ to compute the volume of the solid obtained by revolving the pictured region about the $y$-axis. Can you do it without using a calculator?

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Solution

Solutions---0070-02

03

Area of a garden bed

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Solution

Solutions---0070-03