Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
-The curve $y = e ^ { x }$ and the lines $x + y = 3$ and $x = 3$
A) $\int _ { 0 } ^ { 3 } \int _ { 3 - x } ^ { e ^ { x } } d y d x$
B) $\int _ { 0 } ^ { 3 } \int _ { 3 } ^ { e ^ { x } - x } d y d x$
C) $\int _ { 0 } ^ { 3 } \int _ { \mathrm { x } } ^ { \mathrm { e } ^ { \mathrm { x } } - \mathrm { x } } \mathrm { dydx }$
D) $\int _ { 0 } ^ { 3 } \int _ { x - 3 } ^ { e ^ { x } } d y d x$

Question

Quizplus · Accepted Answer

The correct setup for the iterated double integral involves integrating with respect to $y$ first, from the lower boundary $y = 3 - x$ (derived from the line $x + y = 3$) to the upper boundary $y = e^x$ (the given curve), and then integrating with respect to $x$ from 0 to 3 (since $x = 3$ is a vertical line and the other boundary is the y-axis, which corresponds to $x = 0$).

Express the Area of the Region Bounded by the Given y=exy = e ^ { x }y=ex

Express the Area of the Region Bounded by the Given $y = e ^ { x }$