Question

let $int_{1.5}^{6.5}f(x)dx = 1$, $int_{1.5}^{4}f(x)dx=5$, $int_{4}^{6.5}f(x)dx = 10$. find $int_{4}^{1.5}f(x)dx$ and $int_{1.5}^{4}(1f(x)-5)dx$

Explanation:

Step1: Use integral property

We know that \(\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx\).

Step2: Rewrite the integral

\(\int_{1.5}^{4}f(x)dx=\int_{1.5}^{6.5}f(x)dx-\int_{4}^{6.5}f(x)dx\).

Step3: Substitute given values

Given \(\int_{1.5}^{6.5}f(x)dx = 5\) and \(\int_{4}^{6.5}f(x)dx=10\), then \(\int_{1.5}^{4}f(x)dx=5 - 10=- 5\).

Step4: Calculate \(\int_{1.5}^{4}(1f(x)-5)dx\)

Use the integral - linearity property \(\int_{a}^{b}(cf(x)+d)dx=c\int_{a}^{b}f(x)dx + d(b - a)\), where \(c = 1\), \(d=-5\), \(a = 1.5\), \(b = 4\).
\(\int_{1.5}^{4}(1f(x)-5)dx=\int_{1.5}^{4}f(x)dx-5\int_{1.5}^{4}1dx\).
Since \(\int_{1.5}^{4}1dx=4 - 1.5 = 2.5\) and \(\int_{1.5}^{4}f(x)dx=-5\).
\(\int_{1.5}^{4}(1f(x)-5)dx=-5-5\times2.5=-5 - 12.5=-17.5\).

Answer:

\(-17.5\)