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Math Calculus University Calculus

a. To find: The area under the graph of function by using a lower sum with two rectangles of equal width. The approximate area under the graph of function by using a lower sum with two rectangles of equal width is 0.125 sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for a lower Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 2 = 1 2 Thus, x 1 = 0 , x 2 = 1 2 . ∑ i = 1 n Δ x f x i = 1 2 0 2 + 1 2 2 = 1 2 3 = 1 8 = 0.125 Hence, the area under the graph of function by using a lower sum with two rectangles of equal width is 0.125 sq. units. b. To find: The area under the graph of function by using a lower sum with four rectangles of equal width. The approximate area under the graph of function by using a lower sum with four rectangles of equal width is 0.21875sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for a lower Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 4 = 1 4 Thus, x 1 = 0 , x 2 = 1 4 , x 3 = 1 2 , and x 4 = 3 4 . ∑ i = 1 n Δ x f x i = 1 4 0 2 + 1 4 2 + 1 2 2 + 3 4 2 = 1 4 1 16 + 1 4 + 9 16 = 1 4 14 16 = 7 32 = 0.21875 Hence, the area under the graph of function by using a lower sum with four rectangles of equal width is 0.21875 sq. units. c. To find: The area under the graph of function by using an upper sum with two rectangles of equal width. The approximate area under the graph of function by using an upper sum with two rectangles of equal width is 0.21875sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for an upper Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 2 = 1 2 Thus, x 1 = 1 2 and x 2 = 1 . ∑ i = 1 n Δ x f x i = 1 2 1 2 2 + 1 2 = 1 2 1 4 + 1 = 1 2 5 4 = 5 8 = 0.625 Hence, the area under the graph of function by using an upper sum with two rectangles of equal width is 0.625 sq. units. b. To find: The area under the graph of function by using a lower sum with four rectangles of equal width. The approximate area under the graph of function by using an upper sum with four rectangles of equal width is 0.21875sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for an upper Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 4 = 1 4 Thus, x 1 = 1 4 , x 2 = 1 2 , x 3 = 3 4 , and x 4 = 1 . ∑ i = 1 n Δ x f x i = 1 4 1 4 2 + 1 2 2 + 3 4 2 + 1 2 = 1 4 1 16 + 1 4 + 9 16 + 1 = 1 4 30 16 = 15 32 = 0.46875 Hence, the area under the graph of function by using an upper sum with four rectangles of equal width is 0.46875 sq. units.

a. To find: The area under the graph of function by using a lower sum with two rectangles of equal width. The approximate area under the graph of function by using a lower sum with two rectangles of equal width is 0.125 sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for a lower Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 2 = 1 2 Thus, x 1 = 0 , x 2 = 1 2 . ∑ i = 1 n Δ x f x i = 1 2 0 2 + 1 2 2 = 1 2 3 = 1 8 = 0.125 Hence, the area under the graph of function by using a lower sum with two rectangles of equal width is 0.125 sq. units. b. To find: The area under the graph of function by using a lower sum with four rectangles of equal width. The approximate area under the graph of function by using a lower sum with four rectangles of equal width is 0.21875sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for a lower Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 4 = 1 4 Thus, x 1 = 0 , x 2 = 1 4 , x 3 = 1 2 , and x 4 = 3 4 . ∑ i = 1 n Δ x f x i = 1 4 0 2 + 1 4 2 + 1 2 2 + 3 4 2 = 1 4 1 16 + 1 4 + 9 16 = 1 4 14 16 = 7 32 = 0.21875 Hence, the area under the graph of function by using a lower sum with four rectangles of equal width is 0.21875 sq. units. c. To find: The area under the graph of function by using an upper sum with two rectangles of equal width. The approximate area under the graph of function by using an upper sum with two rectangles of equal width is 0.21875sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for an upper Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 2 = 1 2 Thus, x 1 = 1 2 and x 2 = 1 . ∑ i = 1 n Δ x f x i = 1 2 1 2 2 + 1 2 = 1 2 1 4 + 1 = 1 2 5 4 = 5 8 = 0.625 Hence, the area under the graph of function by using an upper sum with two rectangles of equal width is 0.625 sq. units. b. To find: The area under the graph of function by using a lower sum with four rectangles of equal width. The approximate area under the graph of function by using an upper sum with four rectangles of equal width is 0.21875sq. units. Given information: f x = x 2 between x = 0 and x = 1 . Concept used: The formula for an upper Riemann sum is ∑ i = 1 n Δ f x i , where Δ x = b − a n [ n = Number of rectangles, a = lower limit, b = upper limit ] is the width of each rectangle and f x i gives the height of each rectangle. Explanation: Here, the width of each rectangle will be given by: Δ x = 1 − 0 4 = 1 4 Thus, x 1 = 1 4 , x 2 = 1 2 , x 3 = 3 4 , and x 4 = 1 . ∑ i = 1 n Δ x f x i = 1 4 1 4 2 + 1 2 2 + 3 4 2 + 1 2 = 1 4 1 16 + 1 4 + 9 16 + 1 = 1 4 30 16 = 15 32 = 0.46875 Hence, the area under the graph of function by using an upper sum with four rectangles of equal width is 0.46875 sq. units.

University Calculus

University Calculus

4th Edition

ISBN: 9780135164846

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas, Jr., Przemyslaw Bogacki

Publisher: PEARSON

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