Q.61 Prove Theorem 4.13(c): For any r... [FREE SOLUTION]

Chapter 4: Q.61 (page 354)

Prove Theorem 4.13(c): For any real numbers a and b, ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3}) .$ Use the proof of Theorem 4.13(a) as a guide.

Short Answer

Expert verified

For any real numbers a and b, ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3})$ as follows.

${鈭�}_{a}^{b} x d x = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} {[a + k (\frac{b - a}{n})]}^{2} (\frac{b - a}{n}) = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} a^{2} (\frac{b - a}{n}) + \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} k^{2} {(\frac{b - a}{n})}^{3} + \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} k {(\frac{b - a}{n})}^{2} = \frac{1}{3} (b^{3} - a^{3})$

Step by step solution

Step 1. Given information

The given Integral is ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3}) .$

Step 2. Proof.

Take the interval $[a, b] .$

$鈭� x = \frac{b - a}{n} x_{k} = a + k 鈭� x x_{k} = a + k (\frac{b - a}{n})$

Use the definition of definite integral to find ${鈭�}_{a}^{b} x^{2} d x .$

${鈭�}_{a}^{b} x d x = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} f (\overset{*}{x_{k}}) 鈭� x = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} {[a + k (\frac{b - a}{n})]}^{2} (\frac{b - a}{n}) = \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} a^{2} (\frac{b - a}{n}) + \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} k^{2} {(\frac{b - a}{n})}^{3} + \lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} k {(\frac{b - a}{n})}^{2} = \lim_{n 鈫� 鈭�} a^{2} n (\frac{b - a}{n}) + \lim_{n 鈫� 鈭�} {(\frac{b - a}{n})}^{3} {[\frac{n (n + 1)}{2}]}^{2} + \lim_{n 鈫� 鈭�} {(\frac{b - a}{n})}^{2} [\frac{n (n + 1)}{2}] = \frac{1}{3} (b^{3} - a^{3})$

so ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3})$ for any real number a andb.

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Prove Theorem 4.13(c): For any real numbers a and b, ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3}) .$ Use the proof of Theorem 4.13(a) as a guide.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof.

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91影视

Prove Theorem 4.13(c): For any real numbers a and b, 鈭�abx2dx=13b3-a3.Use the proof of Theorem 4.13(a) as a guide.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof.

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Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Prove Theorem 4.13(c): For any real numbers a and b, ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3}) .$ Use the proof of Theorem 4.13(a) as a guide.