Question

question 7 (1 point)
suppose ( int_{-2}^{7}(3 f(x)) d x=12 ) and ( int_{5}^{7} f(x) d x=2 ). compute ( int_{-2}^{5} f(x) d x ).

view hint for question 7

Explanation:

Step1: Use the property of definite integrals

We know that \(\int_{a}^{b}cf(x)dx = c\int_{a}^{b}f(x)dx\). Given \(\int_{-2}^{7}3f(x)dx = 12\), then \(3\int_{-2}^{7}f(x)dx=12\), so \(\int_{-2}^{7}f(x)dx = 4\).

Step2: Use the additive property of definite integrals

The additive property is \(\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx\). We can write \(\int_{-2}^{7}f(x)dx=\int_{-2}^{5}f(x)dx+\int_{5}^{7}f(x)dx\).

Step3: Substitute the known values

We know that \(\int_{-2}^{7}f(x)dx = 4\) and \(\int_{5}^{7}f(x)dx = 2\). Substituting into \(\int_{-2}^{7}f(x)dx=\int_{-2}^{5}f(x)dx+\int_{5}^{7}f(x)dx\), we get \(4=\int_{-2}^{5}f(x)dx + 2\).

Step4: Solve for \(\int_{-2}^{5}f(x)dx\)

Subtract 2 from both sides of the equation \(4=\int_{-2}^{5}f(x)dx + 2\). So \(\int_{-2}^{5}f(x)dx=4 - 2=2\).

Answer:

\(2\)