Question

given the function graphed below, evaluate the definite integrals.
\\(\int_{0}^{1} f(x) dx = \\)
\\(\int_{1}^{8} f(x) dx = \\)
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Explanation:

Step1: Evaluate $\int_{0}^{1} f(x) dx$

The function $f(x)=2$ on $[0,1]$. The integral is area of a rectangle:
$\int_{0}^{1} f(x) dx = 2 \times (1-0)$

Step2: Calculate first integral result

$\int_{0}^{1} f(x) dx = 2$

Step3: Split $\int_{1}^{8} f(x) dx$ into parts

Split into $\int_{1}^{2} f(x) dx + \int_{2}^{3} f(x) dx + \int_{3}^{8} f(x) dx$

Step4: Evaluate $\int_{1}^{2} f(x) dx$

This is a trapezoid (or triangle+rectangle) with vertices $(1,2),(2,0)$. Area:
$\frac{1}{2} \times (2+0) \times (2-1) = 1$

Step5: Evaluate $\int_{2}^{3} f(x) dx$

This is a trapezoid with vertices $(2,0),(3,-2)$. Area (negative since below x-axis):
$\frac{1}{2} \times (0+(-2)) \times (3-2) = -1$

Step6: Evaluate $\int_{3}^{8} f(x) dx$

$f(x)=-2$ on $[3,8]$. Integral is:
$-2 \times (8-3) = -10$

Step7: Sum the three parts

$\int_{1}^{8} f(x) dx = 1 + (-1) + (-10)$

Answer:

$\int_{0}^{1} f(x) dx = 2$
$\int_{1}^{8} f(x) dx = -10$