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…

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 $int_{1.5}^{4}(1f(x)-5)dx$

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 $int_{1.5}^{4}(1f(x)-5)dx$

Answer

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)