Chapter 6: Q. 1 (page 509)

Calculate each of the following definite integrals, using integration techniques and fundamental theorem of calculus
$\int_{- 3}^{3} (9 - x^{2}) d x \int_{0}^{1} (x^{4} + 2 x^{2}) d x \int_{0}^{1} (4) d x \int_{- 3}^{3} (4 - 9 {(\ln x)}^{2}) d x \int_{0}^{2} (4 y^{2} - y^{4}) d y \int_{0}^{1} (2 - \sqrt{y}) d y$

Short Answer

Expert verified

We have

$\int_{- 3}^{3} (9 - x^{2}) d x = 36 \int_{0}^{1} (x^{4} + 2 x^{2}) d x = \frac{13}{15} \int_{0}^{1} (4) d x = 4 \int_{- 3}^{3} (4 - 9 {(\ln x)}^{2}) d x = - \frac{4}{3} e^{\frac{2}{3}} - 6 \int_{0}^{2} (4 y^{2} - y^{4}) d y = \frac{64}{15} \int_{0}^{1} (2 - \sqrt{y}) d y = \frac{29}{6}$

Step by step solution

Given information

We are given definite integrals we have to evaluate them.

Part a) Step 1: Evaluate

We get,

$\int_{- 3}^{3} (9 - x^{2}) d x = {[9 x - \frac{x^{3}}{3}]}^{3}_{- 3} = [27 - \frac{27}{3}] - [- 27 - (- \frac{27}{3})] = 54 - 18 = 36$

Part b) Step 1: Evaluate

We have,

$\int_{0}^{1} (x^{4} + 2 x^{2}) d x = \frac{x^{5}}{5} + \frac{2 x^{3}}{3} = \frac{1}{5} + \frac{2}{3} = \frac{13}{15}$

Part c) Step 1: Evaluate

We have,

$\int_{0}^{1} 4 d x = {[4 x]}^{1}_{0} = 4$

Part d) Step 1: Evaluate

We have,

$\int_{1}^{e^{\frac{2}{3}}} (4 - 9 {(\ln x)}^{2}) d x P u t \ln x = y h e n c e \frac{1}{x} d x = d y t h e r e f o r e d x = e^{y} d y H e n c e t h e i n t e g r a l b e c o m e s \int_{0}^{\frac{2}{3}} (4 - 9 y^{2}) e^{y} d y = \int_{0}^{\frac{2}{3}} 4 e^{y} - 9 y^{2} e^{y} d y = {[4 e^{y} - 9 y^{2} e^{y} - 2 y e^{y} + 2 e^{y}]}^{\frac{2}{3}}_{0} = - \frac{4}{3} e^{\frac{2}{3}} - 6$

Step e) Step 1: Evaluate

We have,

$\int_{0}^{2} (4 y^{2} - y^{4}) d y = {[\frac{4}{3} y^{3} - \frac{y^{5}}{5}]}^{2}_{0} = \frac{32}{3} - \frac{32}{5} = \frac{64}{15}$

Part f) Step 1: Evaluate

We have,

$\int_{0}^{1} {(2 - \sqrt{y})}^{2} d y = \int_{0}^{1} (4 - 4 \sqrt{y} + y) d y = {[4 y + \frac{2}{3} y^{\frac{2}{3}} + \frac{y^{2}}{2}]}_{0}^{1} = 4 + \frac{2}{3} + \frac{1}{2} = \frac{29}{6}$

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Short Answer

Step by step solution

Given information

Part a) Step 1: Evaluate

Part b) Step 1: Evaluate

Part c) Step 1: Evaluate

Part d) Step 1: Evaluate

Step e) Step 1: Evaluate

Part f) Step 1: Evaluate

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Calculate each of the following definite integrals, using integration techniques and fundamental theorem of calculus∫-33(9-x2)dx∫01(x4+2x2)dx∫01(4)dx∫-33(4-9(lnx)2)dx∫02(4y2-y4)dy∫01(2-y)dy

Short Answer

Step by step solution

Given information

Part a) Step 1: Evaluate

Part b) Step 1: Evaluate

Part c) Step 1: Evaluate

Part d) Step 1: Evaluate

Step e) Step 1: Evaluate

Part f) Step 1: Evaluate

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.