Question

function h is graphed.
find \\(\int_{0}^{4} h(x) \\, dx\\).
\\(\int_{0}^{4} h(x) \\, dx = \square\\)

Explanation:

Step1: Split integral into intervals

$\int_{0}^{4} h(x)dx = \int_{0}^{2} h(x)dx + \int_{2}^{4} h(x)dx$

Step2: Calculate $\int_{0}^{2} h(x)dx$

This is a rectangle with width $2$ and height $2$.
$\int_{0}^{2} h(x)dx = 2 \times 2 = 4$

Step3: Calculate $\int_{2}^{4} h(x)dx$

This is a trapezoid with bases $2$ and $3$, height $2$.
$\int_{2}^{4} h(x)dx = \frac{1}{2} \times (2+3) \times 2 = 5$

Step4: Sum the two results

$\int_{0}^{4} h(x)dx = 4 + 5$

Answer:

$9$