question let f be a continuous function such that ∫−82f(x)dx = 8 and ∫1−8f(x)dx = -3. what is the value of…

Answer:

33

Explanation:

Step1: Use integral property

We know that $\int_{-8}^{2}f(x)dx=\int_{-8}^{1}f(x)dx+\int_{1}^{2}f(x)dx$. Given $\int_{-8}^{2}f(x)dx = 8$ and $\int_{1}^{-8}f(x)dx=- 3$, then $\int_{-8}^{1}f(x)dx = 3$. So, $8 = 3+\int_{1}^{2}f(x)dx$.

Step2: Solve for $\int_{1}^{2}f(x)dx$

$\int_{1}^{2}f(x)dx=8 - 3=5$.

Step3: Use constant - multiple rule of integrals

The constant - multiple rule states that $\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx$ for a constant $k$. For $\int_{1}^{2}3f(x)dx$, with $k = 3$ and $\int_{1}^{2}f(x)dx = 5$, we have $\int_{1}^{2}3f(x)dx=3\times\int_{1}^{2}f(x)dx$.

Step4: Calculate the result

$\int_{1}^{2}3f(x)dx=3\times5 = 33$.