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积分题 10

Easy integration question 10

专题
General / 综合
难度
L4

题目详情

是什么

2∫01/2dx∫1/21/2+xdy?2 \int_{0}^{1 / 2} dx \int_{1 / 2}^{1 / 2 + x} dy?

英文原题

What is

2∫01/2dx∫1/21/2+xdy?2 \int_{0}^{1 / 2} dx \int_{1 / 2}^{1 / 2 + x} dy?
解析

先算内层：

∫1/21/2+xdy=x.\int_{1/2}^{1/2+x}dy=x.

所以原式

2∫01/2x dx=2⋅[x22]01/2=14.2\int_0^{1/2}x\,dx=2\cdot\left[\frac{x^2}{2}\right]_0^{1/2}=\boxed{\frac14}.

英文解析

First, calculate the inner integral:

∫1/21/2+xdy=x.\int_{1/2}^{1/2+x}dy=x.

Thus, the original expression becomes

2∫01/2x dx=2⋅[x22]01/2=14.2\int_0^{1/2}x\,dx=2\cdot\left[\frac{x^2}{2}\right]_0^{1/2}=\boxed{\frac14}.