
Concept explainers
(a) Draw two typical curves y = f(x) and y = g(x), where f(x) ≥ g(x) for a ≤ x ≤ b. Show how to approximate the area between these curves by a Riemann sum and sketch the corresponding approximating rectangles. Then write an expression for the exact area.
(b) Explain how the situation changes if the curves have equations x = f(y) and x = g(y), where f(y) ≥ g(y) for c ≤ y ≤ d.
(a)

To Draw: the two typical curves
To define: A Riemann sum that approximates the area between the two typical curves with drawing of the corresponding approximating rectangles and exact area between the two typical curves and the expression for the exact area.
Explanation of Solution
Consider the two curves
Here, the top curve function is
Assume f and g are continuous function and
Here, the lower limit is a and the upper limit is b.
Show the approximate ith strip rectangle with base
Sketch the two typical curves
Refer to figure 1.
The two typical curves
The expression for the exact area is
Divide the area between the two typical curves into n strips of equal width and take the entire sample points to be right endpoints, in which
Sketch thecorresponding approximating rectangles as shown in Figure 2.
The better and better approximation occurs in
Thus, the Riemann sum with the sketch of corresponding approximating rectangles and the exact area between the two typical curves shown.
Therefore, the approximation of the area between the two typical curves using Riemann sum with the sketch of the corresponding approximating rectangles and the sum of the areas corresponding approximating rectangles is the exact area.
(b)

To Draw: The two typical curves with the changing the situation as
To define: The situation if the curves changes from
The expression for the exact area is
Explanation of Solution
Consider the two curves
Here, the right curve function is
Assume f and g are continuous function and
Here, the bottom limit is c and the top limit is d.
Sketch the two typical curves
Thus, the two typical curves
Normally the height calculated from the top function minus bottom one and integrating from left to right. Instead of normal calculation, use “right minus left” and integrating from bottom to top. Therefore the exact area, A written as
Therefore, the changes of the situation if the curves have equations
Want to see more full solutions like this?
Chapter 6 Solutions
Single Variable Calculus: Early Transcendentals
Additional Math Textbook Solutions
Precalculus: A Unit Circle Approach (3rd Edition)
Pathways To Math Literacy (looseleaf)
Precalculus
Beginning and Intermediate Algebra
Elementary Statistics: Picturing the World (7th Edition)
Calculus: Early Transcendentals (2nd Edition)
- let θ = 17π over 12 Part A: Determine tan θ using the sum formula. Show all necessary work in the calculation.Part B: Determine cos θ using the difference formula. Show all necessary work in the calculation.arrow_forwardCalculus lll May I please have an explanation about how to calculate the derivative of the surface (the dS) on the surface integral, and then explain the essentials of the surface integral?arrow_forwardУ1 = e is a solution to the differential equation xy" — (x+1)y' + y = 0. Use reduction of order to find the solution y(x) corresponding to the initial data y(1) = 1, y′ (1) = 0. Then sin(y(2.89)) is -0.381 0.270 -0.401 0.456 0.952 0.981 -0.152 0.942arrow_forward
- solve pleasearrow_forwardThe parametric equations of the function are given asx=asin²0, y = acos). Calculate [Let: a=anumerical coefficient] dy d²y and dx dx2arrow_forwardA tank contains 200 gal of fresh water. A solution containing 4 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of 5 gal/min. Find the maximum amount of fertilizer in the tank and the time required to reach the maximum. Find the time required to reach the maximum amount of fertilizer in the tank. t= min (Type an integer or decimal rounded to the nearest tenth as needed.)arrow_forward
- Thumbi Irrigation Scheme in Mzimba district is under threat of flooding. In order to mitigate against the problem, authorities have decided to construct a flood protection bund (Dyke). Figure 1 is a cross section of a 300m long proposed dyke; together with its foundation (key). Survey data for the proposed site of the dyke are presented in Table 1. Table 2 provides swelling and shrinkage factors for the fill material that has been proposed. The dyke dimensions that are given are for a compacted fill. (1) Assume you are in the design office, use both the Simpson Rule and Trapezoidal Rule to compute the total volume of earthworks required. (Assume both the dyke and the key will use the same material). (2) If you are a Contractor, how many days will it take to finish hauling the computed earthworks using 3 tippers of 12m³ each? Make appropriate assumptions. DIKE CROSS SECTION OGL KEY (FOUNDATION) 2m 1m 2m 8m Figure 1: Cross section of Dyke and its foundation 1.5m from highest OGL 0.5m…arrow_forwardThe parametric equations of the function are given as x = 3cos 0 - sin³0 and y = 3sin 0 - cos³0. dy d2y Calculate and dx dx².arrow_forward(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z ≤ 3}. Calculate the integral f(x, y, z) dv. Earrow_forward
- (12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.arrow_forward(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.arrow_forward(14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning
- College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning




