current attempt in progress using the figure below, find the value of ∫₅³⁰ f(x)dx. ∫₅³⁰ f(x)dx = etextbook…

Explanation:

Step1: Divide the region into geometric - shapes

The region under the curve from (x = 5) to (x=30) can be divided into 3 trapezoids. The width of each sub - interval (\Delta x=5).

Step2: Recall the area formula for a trapezoid

The area formula of a trapezoid is (A=\frac{1}{2}(b_1 + b_2)h), where (b_1) and (b_2) are the lengths of the parallel sides and (h) is the height.

Step3: Calculate the area of the first trapezoid

For the trapezoid from (x = 5) to (x = 10), (b_1=1), (b_2 = 2), (h = 5). So (A_1=\frac{1}{2}(1 + 2)\times5=\frac{15}{2}).

Step4: Calculate the area of the second trapezoid

For the trapezoid from (x = 10) to (x = 15), (b_1=2), (b_2 = 1), (h = 5). So (A_2=\frac{1}{2}(2 + 1)\times5=\frac{15}{2}).

Step5: Calculate the area of the third trapezoid

For the trapezoid from (x = 15) to (x = 20), (b_1=1), (b_2 = 2), (h = 5). So (A_3=\frac{1}{2}(1 + 2)\times5=\frac{15}{2}).

Step6: Calculate the definite integral

(\int_{5}^{30}f(x)dx=A_1+A_2+A_3=\frac{15}{2}+\frac{15}{2}+\frac{15}{2}=\frac{45}{2}=22.5)

Answer:

(22.5)