Math / Calculus

Question ${ }_{1}^{8} 3$

Studdy Solution

STEP 1

1. We are given two separate double integral problems.
2. The first problem involves evaluating the integral of $2x$ over a rectangular region $R$ .
3. The second problem involves evaluating the integral of $2$ over a triangular region $D$ .

STEP 2

1. Evaluate the first double integral over region $R$ .
2. Evaluate the second double integral over region $D$ .

STEP 3

Identify the limits of integration for the region $R$ defined by $0 \leq x \leq 2$ and $1 \leq y \leq 2$ .

STEP 4

Set up the double integral for the first problem:
$\int_{1}^{2} \int_{0}^{2} 2x \, dx \, dy$

STEP 5

Evaluate the inner integral with respect to $x$ :
$\int_{0}^{2} 2x \, dx = \left[ x^2 \right]_{0}^{2} = 4 - 0 = 4$

STEP 6

Evaluate the outer integral with respect to $y$ :
$\int_{1}^{2} 4 \, dy = 4y \bigg|_{1}^{2} = 4(2) - 4(1) = 8 - 4 = 4$
The answer to the first problem is $\boxed{4}$ .

STEP 7

Identify the limits of integration for the region $D$ defined by $0 \leq x \leq 1$ and $x \leq y \leq 1 + x$ .

STEP 8

Set up the double integral for the second problem:
$\int_{0}^{1} \int_{x}^{1+x} 2 \, dy \, dx$

STEP 9

Evaluate the inner integral with respect to $y$ :
$\int_{x}^{1+x} 2 \, dy = 2y \bigg|_{x}^{1+x} = 2(1+x) - 2(x) = 2 + 2x - 2x = 2$

STEP 10

Evaluate the outer integral with respect to $x$ :
$\int_{0}^{1} 2 \, dx = 2x \bigg|_{0}^{1} = 2(1) - 2(0) = 2$
The answer to the second problem is $\boxed{2}$ .

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