the graph h is shown below. evaluate the definite integral of ∫−17−9h(x)dx.
the graph h is shown below. evaluate the definite integral of ∫−17−9h(x)dx.
Answer
Explanation:
Step1: Divide the region into shapes
The region under the curve from $x = - 17$ to $x=-9$ can be divided into a trapezoid and a triangle.
Step2: Calculate area of trapezoid
The formula for the area of a trapezoid is $A_1=\frac{1}{2}(b_1 + b_2)h$. For the trapezoid from $x=-17$ to $x = - 13$, $b_1=-2$, $b_2 = 0$, $h = 4$. So $A_1=\frac{1}{2}(-2+0)\times4=-4$.
Step3: Calculate area of triangle
The formula for the area of a triangle is $A_2=\frac{1}{2}bh$. For the triangle from $x=-13$ to $x=-9$, $b = 4$, $h=4$. So $A_2=\frac{1}{2}\times4\times4 = 8$.
Step4: Sum the areas
The value of the definite - integral $\int_{-17}^{-9}h(x)dx$ is the sum of the areas of the trapezoid and the triangle. So $\int_{-17}^{-9}h(x)dx=A_1 + A_2=-4 + 8=4$.
Answer:
$4$