Chapter 13: Q. 1 (page 1082)

Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let $R = \{(x, y) | 0 \leq x \leq 2 a n d 1 \leq y \leq 4\}$ .
$\iint (x + 2 y) d A$

Short Answer

Expert verified

$\iint (x + 2 y) d A = 36 s q u a r e u n i t s$

Step by step solution

Definition of double integral

Let $R = \{(x, y) | a \leq x \leq b a n d c \leq y \leq d\}$ and $f (x, y)$ is a continuous function on R, Then

$\iint f (x, y) d A = \int_{a}^{b} \int_{c}^{d} f (x, y) d x d y$

Evaluate the integral

$\iint (x + 2 y) d A = \int_{1}^{4} \int_{0}^{2} (x + 2 y) d x d y = \int_{1}^{4} [\int_{0}^{2} (x + 2 y) d x] d y = \int_{1}^{4} d y {[\frac{x^{2}}{2} + 2 x y]}_{0}^{2} = \int_{1}^{4} d y [\frac{2^{2}}{2} + 2 \times 2 \times y] = \int_{1}^{4} (2 + 4 y) d y = {[2 y + 4 \frac{y^{2}}{2}]}_{1}^{4} = (8 + 32) - (2 + 2) = 40 - 4 = 36$

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Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let $R = \{(x, y) | 0 \leq x \leq 2 a n d 1 \leq y \leq 4\}$ .
$\iint (x + 2 y) d A$

Short Answer

Step by step solution

Definition of double integral

Evaluate the integral

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Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let R=(x,y)|0≤x≤2and1≤y≤4.∬(x+2y)dA

Short Answer

Step by step solution

Definition of double integral

Evaluate the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Pure Maths

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Using the definition to evaluate a double integral: Evaluate the given double integrals as a limit of a Riemann sum. For each integral, let $R = \{(x, y) | 0 \leq x \leq 2 a n d 1 \leq y \leq 4\}$ .
$\iint (x + 2 y) d A$