Use finite approximation to estimate the area under the graph of f(x) = 4x² and above the graph of f(x) = 0 from x = 0 to x₁ = 10 using i) a lower sum with two rectangles of equal width. ii) a lower sum with four rectangles of equal width. iii) an upper sum with two rectangles of equal width. iv) an upper sum with four rectangles of equal width. The estimated area using a lower sum with two rectangles of equal width is (Simplify your answer. Type an integer or a decimal.) square units.
Use finite approximation to estimate the area under the graph of f(x) = 4x² and above the graph of f(x) = 0 from x = 0 to x₁ = 10 using i) a lower sum with two rectangles of equal width. ii) a lower sum with four rectangles of equal width. iii) an upper sum with two rectangles of equal width. iv) an upper sum with four rectangles of equal width. The estimated area using a lower sum with two rectangles of equal width is (Simplify your answer. Type an integer or a decimal.) square units.
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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Transcribed Image Text:**Problem Statement:**
Use finite approximation to estimate the area under the graph of \( f(x) = 4x^2 \) and above the graph of \( f(x) = 0 \) from \( x_0 = 0 \) to \( x_n = 10 \) using:
i) A lower sum with two rectangles of equal width.
ii) A lower sum with four rectangles of equal width.
iii) An upper sum with two rectangles of equal width.
iv) An upper sum with four rectangles of equal width.
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**Calculation Prompt:**
The estimated area using a lower sum with two rectangles of equal width is \(\_\_\_\) square units.
*(Simplify your answer. Type an integer or a decimal.)*
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**Explanation for Educational Insight:**
This problem involves estimating the area under a curve using Riemann sums, which is a common method in calculus to approximate integrals. The lower sum uses the smallest function values in each subinterval to form rectangles, while the upper sum uses the largest. Adjusting the number of rectangles affects the accuracy of the approximation.
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