Find the area of the region specified by the integral(s).
-$$\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 9 - x } d y d x$$
A) $\frac { 81 } { 4 }$
B) 27
C) $\frac { 81 } { 2 }$
D) $\frac { 27 } { 2 }$

Question

Quizplus · Accepted Answer

We have to find the area of the region under $x+y=9$ and above the x and y axis, which is a triangle. We can rewrite the integral bounds as $\int_0^9\int_0^{9-x}dydx$ or $\int_0^9\int_0^{9-y}dxdy$ (both represent the same region). Evaluating either integral using Fubini's Theorem gives us $\frac{81}{2}$.

Find the Area of the Region Specified by the Integral(s) ∫09∫09−xdydx\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 9 - x } d y d x∫09​∫09−x​dydx

Find the Area of the Region Specified by the Integral(s) $\int _ { 0 } ^ { 9 } \int _ { 0 } ^ { 9 - x } d y d x$