Question

use these values to evaluate the given definite integrals.
\\( \int _ { 0 } ^ { 2 } f ( x ) d x = - 10, \quad \int _ { 0 } ^ { 2 } g ( x ) d x = 8, \quad \int _ { 0 } ^ { 3 } f ( x ) d x = 6, \quad \int _ { 2 } ^ { 3 } g ( x ) d x = 2 \\)
\\( \

$$\begin{array} { l } { \\text { a. } \\int _ { 0 } ^ { 3 } ( f ( x ) + g ( x ) ) d x = } \\\\ { \\text { b. } \\int _ { 0 } ^ { 3 } ( f ( x ) - g ( x ) ) d x = } \\\\ { \\text { c. } \\int _ { 2 } ^ { 3 } ( 3 f ( x ) + 2 g ( x ) ) d x = } \\\\ { \\text { d. find the value } a \\text { such that } } \\\\ { \\quad \\int _ { 0 } ^ { 3 } ( a f ( x ) + g ( x ) ) d x = 0. } \\\\ { a = } \\end{array}$$

\\)

Explanation:

Step1: Split the integral

$\int_{0}^{2} [f(x) + g(x)]dx = \int_{0}^{2}f(x)dx + \int_{0}^{2}g(x)dx$

Step2: Substitute known values

$= -10 + 8 = -2$

Step3: Split the integral

$\int_{0}^{3} [f(x) - g(x)]dx = \int_{0}^{3}f(x)dx - \int_{0}^{3}g(x)dx$

Step4: Compute $\int_{0}^{3}g(x)dx$

$\int_{0}^{3}g(x)dx = \int_{0}^{2}g(x)dx + \int_{2}^{3}g(x)dx = 8 + 2 = 10$

Step5: Substitute values

$= 6 - 10 = -4$

Step6: Split the integral

$\int_{2}^{3} [3f(x) + 2g(x)]dx = 3\int_{2}^{3}f(x)dx + 2\int_{2}^{3}g(x)dx$

Step7: Compute $\int_{2}^{3}f(x)dx$

$\int_{2}^{3}f(x)dx = \int_{0}^{3}f(x)dx - \int_{0}^{2}f(x)dx = 6 - (-10) = 16$

Step8: Substitute values

$= 3(16) + 2(2) = 48 + 4 = 52$

Step9: Split the integral

$\int_{0}^{3} [af(x) + g(x)]dx = a\int_{0}^{3}f(x)dx + \int_{0}^{3}g(x)dx = 0$

Step10: Solve for $a$

$6a + 10 = 0 \implies 6a = -10 \implies a = -\frac{10}{6} = -\frac{5}{3}$

Answer:

-2
-4
52
$-\frac{5}{3}$