Explanation Given information: y = 3 x 1 − x Formula: ( a − b ) 2 = a 2 + b 2 − 2 a b ( a + b ) 2 = a 2 + b 2 + 2 a b a 2 − b 2 = ( a + b ) ( a − b ) ∫ a b x n d x = x n + 1 n + 1 Calculation: Express the integral as below. y = 3 x 1 − x Show the curve for the given function as in Figure (1). Assume the radius of the shell as x and height of the shell as y . Calculate the volume of the solid using the given expression. V = ∫ a b 2 π ( Radius of shell ) ( Height of shell ) d x Substitute x for radius of shell and y for height of shell. V = ∫ a b 2 π x y d x Substitute 3 x 1 − x for y . V = ∫ a b 2 π x ( 3 x 1 − x ) d x = ∫ a b 6 π x 2 1 − x d x V = 6 π ∫ a b x 2 1 − x d x (1) Apply integral substitution by assuming u as 1 − x . u = 1 − x Differentiate with respect to x . x = 1 − u d u d x = − 1 − d u = d x For the variable x , the interval lies between a to b . For the variable u , the interval lies between 0 to 1. Substitute u for 1 − x , 1 − u for x and − d u for d x in the equation (1)

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

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

Publisher: PEARSON

See similar textbooks

Concept explainers

Cylinders

A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.

Cones

A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is dr…

Videos

Question

Chapter 8, Problem 13AAE

To determine

The volume of solid.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8, Problem 12AAE

Chapter 8, Problem 14AAE

Students have asked these similar questions

The parametric equations of the function are given asx=asin²0, y = acos). Calculate [Let: a=anumerical coefficient] dy d²y and dx dx2

A 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.)

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…

Chapter 8 Solutions

University Calculus: Early Transcendentals (4th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.