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Step 1 Take the direct antiderivative using the Power Rule for Integration. (x + 2)² When n-4. Ar Stap 3 (x + 2)² Stap 2 Apply the Trapezoidal Rule, let n-4. Step 5 Ax- As n 4 ✓ (x + 2)² dx- F(x) dx -✓ 3 ₁-[ = 1 Substitute these values in the Trapezoidal Rule. 2 dx = Here a 3x X 0.3333✔ X 24 1/4 Ax- 1/2 ✔ 3/4 (0✔ (1/2 , therefore the corresponding x values are the following. (3/4✔ (✔ 3352 ✔ Therefore, we have the following. 2 25 121 9 0.3333 (rounded to four decimal places) 2 Therefore, the value of the given integral obtained by using the Trapezoidal Rule is 3352 ✔ + Stap 4 Now use Simpson's Rule, which states the following. Tole(x) ](x) - (₂) (...)+(x) 1/4 + 2)2 +2) 1/2 + 2)² 3/4 3/4 +212 1+2) 32 32 0.3352 (rounded to four decimal places) 1x x and fx) - 2 0.3352

Step 1 Take the direct antiderivative using the Power Rule for Integration. (x + 2)² When n-4. Ar Stap 3 (x + 2)² Stap 2 Apply the Trapezoidal Rule, let n-4. Step 5 Ax- As n 4 ✓ (x + 2)² dx- F(x) dx -✓ 3 ₁-[ = 1 Substitute these values in the Trapezoidal Rule. 2 dx = Here a 3x X 0.3333✔ X 24 1/4 Ax- 1/2 ✔ 3/4 (0✔ (1/2 , therefore the corresponding x values are the following. (3/4✔ (✔ 3352 ✔ Therefore, we have the following. 2 25 121 9 0.3333 (rounded to four decimal places) 2 Therefore, the value of the given integral obtained by using the Trapezoidal Rule is 3352 ✔ + Stap 4 Now use Simpson's Rule, which states the following. Tole(x) ](x) - (₂) (...)+(x) 1/4 + 2)2 +2) 1/2 + 2)² 3/4 3/4 +212 1+2) 32 32 0.3352 (rounded to four decimal places) 1x x and fx) - 2 0.3352

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. dx, n-4 So (x + 2)² Step 1 Take the direct antiderivative using the Power Rule for Integration. 2 When n = 4, Ax- f (x + 2)² Step 3 Step 2 Apply the Trapezoidal Rule, let n 4. Step 5 1/4✔ 1/2✔ 4. 3/4✔ As - dx- Jo x + 2)² x #0 ₂-0 1 dx = X₂-1 Here a-3× F 3✔ Substitute these values in the Trapezoidal Rule. 2 + Ax 0.3333 0.3333✔ 1/4 1/2 3/4 ✓ +2 ✔ 2 do✔ (1/4✔ , therefore the corresponding x values are the following. 4 1/2✔ (3/4✔ 3352 32✔ 3 25 Therefore, we have the following. xo- X₁- 81 3 121 (+2)² 9 n 0.3333 (rounded to four decimal places) 2 0+2) + 2 Therefore, the value of the given integral obtained by using the Trapezoidal Rule is.3352✔ Step 4 Now use Simpson's Rule, which states the following 1/4 [**xxx3[*(x) 1/2 + 2)2 3/4 +212 32 32 0.3352 (rounded to four decimal places) +3× 4(x) + ✓ Zr(₂) [4] (x,-1) + (x₂)] x n-3x and x) - (x + 2)² 0.3352
Transcribed Image Text:Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of n. Round your answer to four decimal places and compare the results with the exact value of the definite integral. dx, n-4 So (x + 2)² Step 1 Take the direct antiderivative using the Power Rule for Integration. 2 When n = 4, Ax- f (x + 2)² Step 3 Step 2 Apply the Trapezoidal Rule, let n 4. Step 5 1/4✔ 1/2✔ 4. 3/4✔ As - dx- Jo x + 2)² x #0 ₂-0 1 dx = X₂-1 Here a-3× F 3✔ Substitute these values in the Trapezoidal Rule. 2 + Ax 0.3333 0.3333✔ 1/4 1/2 3/4 ✓ +2 ✔ 2 do✔ (1/4✔ , therefore the corresponding x values are the following. 4 1/2✔ (3/4✔ 3352 32✔ 3 25 Therefore, we have the following. xo- X₁- 81 3 121 (+2)² 9 n 0.3333 (rounded to four decimal places) 2 0+2) + 2 Therefore, the value of the given integral obtained by using the Trapezoidal Rule is.3352✔ Step 4 Now use Simpson's Rule, which states the following 1/4 [**xxx3[*(x) 1/2 + 2)2 3/4 +212 32 32 0.3352 (rounded to four decimal places) +3× 4(x) + ✓ Zr(₂) [4] (x,-1) + (x₂)] x n-3x and x) - (x + 2)² 0.3352
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