Question

Given the integral:

∫ ʸ=¹/y=₀​ ∫x=/ √y​ x=1 eˣ³ dxdy ​​​​​​​
a) Sketch the integration region of the integral.
b) Change the integration order and calculate the integral.

Answer 1

To address the student's question, we first sketch the integration region for the given integral, then change the integration order, and finally complete the calculation with the new limits.

Step-by-step explanation:

The student asked to sketch the integration region and change the order of integration for the given integral: ∫ y=1 to y=0 ∫ x=1 to x=√ y e³ dx dy. The correct approach involves two main steps:

1. Sketch the integration region: This would be the area in the xy-plane bounded by x=1, x=√ y, y=0, and y=1.
2. Change the integration order and calculate the integral: We would first find the new limits of integration for x and y based on the sketched region and then perform the integral with the new order.

Sketching the region, we would find that x varies from √ y to 1 as y goes from 0 to 1. To change the integration order, we need to express y in terms of x. Since x=√ y, then y=x². This implies that x varies between 0 and 1, and for each x, y varies from 0 to x². Thus, the integral with switched limits becomes ∫ x=0 to x=1 ∫ y=0 to y=x² e³ dy dx, which can then be solved.