Find the area of the region specified by the integral(s).
-$$\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { y } d x d y + \int _ { 4 } ^ { 8 } \int _ { 0 } ^ { 8 - y } d x d y$$
A) 32
B) 2
C) 16
D) 4

Question

Quizplus · Accepted Answer

The given integral represents a region which is a right-angled triangle with vertices (0,0), (4,4) and (8,0). To find the area of this triangle, we can divide it into two triangles using the line x=4, and then add the areas of the two triangles.

The first integral represents the region in the triangle with vertices (0,0), (4,4) and (4,0). This is a right-angled triangle with base 4 and height 4, so its area is (1/2)(4)(4) = 8.

The second integral represents the region in the triangle with vertices (4,4), (8,0) and (8,4). This is also a right-angled triangle with base 4 and height 4, so its area is (1/2)(4)(4) = 8.

Therefore, the total area of the region is 8+8 = 16.

Hence, the answer is C.

Find the Area of the Region Specified by the Integral(s) ∫04∫0ydxdy+∫48∫08−ydxdy\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { y } d x d y + \int _ { 4 } ^ { 8 } \int _ { 0 } ^ { 8 - y } d x d y∫04​∫0y​dxdy+∫48​∫08−y​dxdy

Find the Area of the Region Specified by the Integral(s) $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { y } d x d y + \int _ { 4 } ^ { 8 } \int _ { 0 } ^ { 8 - y } d x d y$