Chapter 13: Q. 42 (page 1027)

Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in $ℝ^{3}$ . Use polar coordinates to describe the solid, and evaluate the expressions.
$\int_{0}^{π} \int_{0}^{R} (R^{2} r - r^{3}) d r d θ$

Short Answer

Expert verified

The value of integral is $\int_{0}^{z} \int_{0}^{π} (R^{2} r - r^{3}) d r d θ = \frac{π R^{4}}{4}$

Step by step solution

Given information

The expression is $I = \int_{0}^{π} \int_{0}^{R} (R^{2} r - r^{3}) d r d θ$

Calculation

The objective of this problem is to use polar coordinates to describe the solid and evaluate the integral expression.

$I = \int_{0}^{π} \int_{0}^{R} (R^{2} r - r^{3}) d r d θ$

Here, $r = 0, r = R$ and $θ = 0, θ = π$

Integrate with respect to r first,

$I = \int_{0}^{π} {[\frac{R^{2} r^{2}}{2} - \frac{r^{4}}{4}]}_{0}^{R} d θ [\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C]$

Substitute the limits

$I = \int_{0}^{π} [\frac{R^{4}}{2} - \frac{R^{4}}{4}] d θ$

I = i n t_{0}^{π} [\frac{R^{4}}{4}] d θ

Now integrate with respect to $θ$

I = \frac{R^{4}}{4} [θ]_{0}^{π} I = \frac{π R^{4}}{4}

Thus, the value of integral is

$\int_{0}^{z} \int_{0}^{π} (R^{2} r - r^{3}) d r d θ = \frac{π R^{4}}{4}$

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Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in $ℝ^{3}$ . Use polar coordinates to describe the solid, and evaluate the expressions.
$\int_{0}^{π} \int_{0}^{R} (R^{2} r - r^{3}) d r d θ$

Short Answer

Step by step solution

Given information

Calculation

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Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in ℝ3. Use polar coordinates to describe the solid, and evaluate the expressions.∫0π∫0RR2r-r3drdθ

Short Answer

Step by step solution

Given information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

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Calculus

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Study anywhere. Anytime. Across all devices.

Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in $ℝ^{3}$ . Use polar coordinates to describe the solid, and evaluate the expressions.
$\int_{0}^{π} \int_{0}^{R} (R^{2} r - r^{3}) d r d θ$