Chapter 5: Q. 3 (page 477)

The Fundamental Theorem of Calculus: Solve each of the following definite integrals.
$\int_{1}^{100} \frac{1}{\sqrt[3]{x^{4}}} d x$ .
$\int_{0.01}^{1} \frac{1}{\sqrt[3]{x^{4}}} d x$ .
$\int_{1}^{100} \frac{1}{\sqrt[4]{x^{3}}} d x$ .
$\int_{0.01}^{1} \frac{1}{\sqrt[4]{x^{3}}} d x$ .

Short Answer

Expert verified

The solution for the given integrals are,

Part $(a)$ : role="math" localid="1649140409260" $\int_{1}^{100} \frac{1}{\sqrt[3]{x^{4}}} d x \approx 2.3537$ .

Part $(b)$ : $\int_{0.01}^{1} \frac{1}{\sqrt[3]{x^{4}}} d x \approx 10.9248$ .

Part $(c)$ : $\int_{1}^{100} \frac{1}{\sqrt[4]{x^{3}}} d x \approx 2.1623$ .

Part $(d)$ : $\int_{0.01}^{1} \frac{1}{\sqrt[4]{x^{3}}} d x \approx 0.6838$ .

Step by step solution

Part a Step 1. Given information

$\int_{1}^{100} \frac{1}{\sqrt[3]{x^{4}}} d x$ .

Part a Step 2. Now apply the fundamental theorem of calculus in the given integral

$\int_{1}^{100} \frac{1}{\sqrt[3]{x^{4}}} d x = \int_{1}^{100} \frac{1}{{(x^{4})}^{\frac{1}{3}}} d x = \int_{1}^{100} {(x^{4})}^{- \frac{1}{3}} d x = {[\frac{x^{\frac{- 1}{3}}}{\frac{- 1}{3}}]}_{1}^{100} = - 3 [{(100)}^{\frac{- 1}{3}} - 1] = - 3 (0.2154 - 1) \approx 2.3537$

Part b Step 1. Given information

$\int_{0.01}^{1} \frac{1}{\sqrt[3]{x^{4}}} d x$ .

Part b Step 2. Now apply the fundamental theorem of calculus in the given integral

$\int_{0.01}^{1} \frac{1}{\sqrt[3]{x^{4}}} d x = - 3 {[x^{\frac{- 1}{3}}]}_{0.01}^{1} = - 3 [1 - 4.6416] \approx 10.9248$

Part c Step 1. Given information

$\int_{1}^{100} \frac{1}{\sqrt[4]{x^{3}}} d x$ .

Part c Now apply the fundamental theorem of calculus in the given interval.

$\int_{1}^{100} \frac{1}{\sqrt[4]{x^{3}}} d x = {[x^{\frac{1}{4}}]}_{1}^{100} = [{(10^{2})}^{\frac{1}{4}} - 1] \approx 2.1623$

Part d Step 1. Given information

$\int_{0.01}^{1} \frac{1}{\sqrt[4]{x^{3}}} d x$ .

Part d Step 2. Now apply the fundamental theorem of calculus in the given interval.

$\int_{0.01}^{1} \frac{1}{\sqrt[4]{x^{3}}} d x = {[x^{\frac{1}{4}}]}_{0.01}^{1} = 1 - {(1 \times 10^{- 2})}^{\frac{1}{4}} \approx 0.6838$

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The Fundamental Theorem of Calculus: Solve each of the following definite integrals.
$\int_{1}^{100} \frac{1}{\sqrt[3]{x^{4}}} d x$ .
$\int_{0.01}^{1} \frac{1}{\sqrt[3]{x^{4}}} d x$ .
$\int_{1}^{100} \frac{1}{\sqrt[4]{x^{3}}} d x$ .
$\int_{0.01}^{1} \frac{1}{\sqrt[4]{x^{3}}} d x$ .

Short Answer

Step by step solution

Part a Step 1. Given information

Part a Step 2. Now apply the fundamental theorem of calculus in the given integral

Part b Step 1. Given information

Part b Step 2. Now apply the fundamental theorem of calculus in the given integral

Part c Step 1. Given information

Part c Now apply the fundamental theorem of calculus in the given interval.

Part d Step 1. Given information

Part d Step 2. Now apply the fundamental theorem of calculus in the given interval.

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The Fundamental Theorem of Calculus: Solve each of the following definite integrals.∫11001x43dx.∫0.0111x43dx.∫11001x34dx.∫0.0111x34dx.

Short Answer

Step by step solution

Part a Step 1. Given information

Part a Step 2. Now apply the fundamental theorem of calculus in the given integral

Part b Step 1. Given information

Part b Step 2. Now apply the fundamental theorem of calculus in the given integral

Part c Step 1. Given information

Part c Now apply the fundamental theorem of calculus in the given interval.

Part d Step 1. Given information

Part d Step 2. Now apply the fundamental theorem of calculus in the given interval.

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The Fundamental Theorem of Calculus: Solve each of the following definite integrals.
$\int_{1}^{100} \frac{1}{\sqrt[3]{x^{4}}} d x$ .
$\int_{0.01}^{1} \frac{1}{\sqrt[3]{x^{4}}} d x$ .
$\int_{1}^{100} \frac{1}{\sqrt[4]{x^{3}}} d x$ .
$\int_{0.01}^{1} \frac{1}{\sqrt[4]{x^{3}}} d x$ .