Suppose $X_1$ and $X_2$ are two independent random variables that follow a uniform distribution and $0\leq x_1 \leq k$, $0\leq x_2 \leq k$, where $k$ is a positive constant. Now I want to find the expectation of the function $f_{X_2}(a+bx_1)$ in the region $x_1 \leq a$, i.e., $$ \int_{0}^{a} f_{X_2}(a+bx_1)f(x_1)dx_1$$
Here is my work: $$\int_{0}^{a}f_{X_2}(a+bx_1)f(x_1)dx_1=\int_{0}^{a}\frac{1}{(a+bk)-(a+b*0)}f(x_1)dx_1\\=\int_{0}^{a}\frac{1}{bk}f(x_1)dx_1=\frac{1}{bk} \frac{a-0}{k-0}=\frac{1}{bk} \frac{a}{k}$$ Is this the right approach?
Thank you!