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 $\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx$. So, $\int_{-8}^{2}f(x)dx=\int_{-8}^{1}f(x)dx+\int_{1}^{2}f(x)dx$. 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, $\int_{1}^{2}f(x)dx=\int_{-8}^{2}f(x)dx-\int_{-8}^{1}f(x)dx=8 - 3=5$.

Step2: Use constant - multiple rule of integration

The constant - multiple rule of integration states that $\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx$ for a constant $k$. Here $k = 3$ and we want to find $\int_{1}^{2}3f(x)dx$. Since $\int_{1}^{2}f(x)dx = 5$, then $\int_{1}^{2}3f(x)dx=3\int_{1}^{2}f(x)dx$.

Step3: Calculate the result

Substitute $\int_{1}^{2}f(x)dx = 5$ into $3\int_{1}^{2}f(x)dx$. We get $3\times5 = 33$.