Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
-The lines x = 0, y = 3x, and y = 7
A) $\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { y / 7 } \mathrm { dx } \mathrm { dy }$
B) $\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { x / 7 } \mathrm { dydx }$
C) $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { y / 3 } \mathrm { dx } \mathrm { dy }$
D) $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { y / 3 } \mathrm { dydx }$

Question

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
-The lines x = 0, y = 3x, and y = 7 
A) $\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { y / 7 } \mathrm { dx } \mathrm { dy }$
B) $\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { x / 7 } \mathrm { dydx }$
C) $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { y / 3 } \mathrm { dx } \mathrm { dy }$
D) $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { y / 3 } \mathrm { dydx }$

Quizplus · Accepted Answer

The region is bounded by x = 0 (a vertical line), y = 3x (a line passing through the origin with a slope of 3), and y = 7 (a horizontal line). To express the area as an iterated double integral, we first fix y between its lower and upper bounds (0 and 7, respectively). For each fixed y, x varies from 0 to y/3 (since y = 3x, x = y/3). Thus, the correct expression is $\int _ { 0 } ^ { 7 } \int _ { 0 } ^ { y / 3 } \mathrm { dx } \mathrm { dy }$.

Express the Area of the Region Bounded by the Given ∫03∫0y/7dxdy\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { y / 7 } \mathrm { dx } \mathrm { dy }∫03​∫0y/7​dxdy

Express the Area of the Region Bounded by the Given $\int _ { 0 } ^ { 3 } \int _ { 0 } ^ { y / 7 } \mathrm { dx } \mathrm { dy }$