this question is designed to be answered without a calculator. if f and g are functions such that…

this question is designed to be answered without a calculator. if f and g are functions such that $int_{0}^{2}f(x)dx = 2$ and $int_{0}^{2}(f(x)-2g(x))dx = 8$, what is the value of $int_{0}^{2}g(x)dx$? -12 -3 3 12

this question is designed to be answered without a calculator. if f and g are functions such that $int_{0}^{2}f(x)dx = 2$ and $int_{0}^{2}(f(x)-2g(x))dx = 8$, what is the value of $int_{0}^{2}g(x)dx$? -12 -3 3 12

Answer

Explanation:

Step1: Expand the integral

By the property of definite - integrals $\int_{a}^{b}(u(x)-v(x))dx=\int_{a}^{b}u(x)dx-\int_{a}^{b}v(x)dx$. So, $\int_{0}^{2}(f(x)-2g(x))dx=\int_{0}^{2}f(x)dx - 2\int_{0}^{2}g(x)dx$.

Step2: Substitute the given values

We know that $\int_{0}^{2}f(x)dx = 2$ and $\int_{0}^{2}(f(x)-2g(x))dx = 8$. Substitute these values into the equation from Step 1: $8=2-2\int_{0}^{2}g(x)dx$.

Step3: Solve for $\int_{0}^{2}g(x)dx$

First, subtract 2 from both sides of the equation: $8 - 2=-2\int_{0}^{2}g(x)dx$, which simplifies to $6=-2\int_{0}^{2}g(x)dx$. Then divide both sides by - 2: $\int_{0}^{2}g(x)dx=-3$.

Answer:

-3