Evaluate the double integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$ .
A)134.00
B)144.00
C)154.00
D)42.00
E)24.00

Question

Quizplus · Accepted Answer

The given double integral can be solved by first integrating with respect to $y$ and then with respect to $x$. Integrating $3x + 4y$ with respect to $y$ from 0 to 3 gives $3xy + 2y^2$ evaluated from 0 to 3, which simplifies to $9x + 18$. Then, integrating $9x + 18$ with respect to $x$ from 0 to 4 gives $4.5x^2 + 18x$ evaluated from 0 to 4, which simplifies to $72$. Therefore, the value of the double integral is 72, but since the integration with respect to $y$ was not correctly evaluated in this explanation, the correct procedure is as follows:First, integrate $3x + 4y$ with respect to $y$: $\int_{0}^{3} (3x + 4y) dy = 3x*y + 2y^2 |_{0}^{3} = 9x + 18$.Then, integrate the result with respect to $x$: $\int_{0}^{4} (9x + 18) dx = 4.5x^2 + 18x |_{0}^{4} = 72 + 72 = 144$.So, the correct answer is 144.

Evaluate the Double Integral ∫04∫03(3x+4y)dydx\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x∫04​∫03​(3x+4y)dydx

Evaluate the Double Integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$