Q. 1 Calculate each of the following ... [FREE SOLUTION]

Chapter 6: Q. 1 (page 509)

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

Short Answer

Expert verified

We have

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

Step by step solution

Given information

We are given definite integrals we have to evaluate them.

Part a) Step 1: Evaluate

We get,

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

Part b) Step 1: Evaluate

We have,

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

Part c) Step 1: Evaluate

We have,

${鈭�}_{0}^{1} 4 d x = {[4 x]}^{1}_{0} = 4$

Part d) Step 1: Evaluate

We have,

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

Step e) Step 1: Evaluate

We have,

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

Part f) Step 1: Evaluate

We have,

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

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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鈭�-33(9-x2)dx鈭�01(x4+2x2)dx鈭�01(4)dx鈭�-33(4-9(lnx)2)dx鈭�02(4y2-y4)dy鈭�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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

Mechanics Maths

Pure Maths

Study anywhere. Anytime. Across all devices.