Exercise (b)the centroid of the region bounded by the graphs of the equationsStep 1The region is bounded by the graphs of the equationsy sin x, y = 0, x = 0, and xx.The area of the representative rectangle isdA= (sin xSo, the area of the entire region isA ==Therefore,LetLetFor the above region,f(x) = sin xg(x) = 0a = 0,b=y=sin(x)KU=2Use integration by parts.Step 2The x-coordinate of the centroid for a region of constant density isx = = ®x[f(x) - g(x)] ddx.-cos xV =-1Therefore,Differentiate with respect to x on both sides.du = dxUsing integration by partsJu dv = uv- -/₁dv sin x dx.Integrate with respect to x on both sides.x=1* = = 6*²*x-1/[x²x sin x dxv du.sin (r)) dx.✓sin x dx-x cos x +2- cos (r)+6²°-x cos x +x sin x dx.- 1/21 - x0The x-coordinate of the centroid of the region is1cos x dx10

Answered: Exercise (b) the centroid of the region…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Similar questions

volume of the solid generated when the region bounded by y = 9 − x2 and y = 2x + 6 is revolved about the x-axis.
The area of the region is bounded by the x-axis, the curve y = 6x – x2, the line x – 1 = 0 and the line x = 4. 3. Find the ordinate of the vertex of the curve y = 6x – x2. Find the area of the region. Find the distance of the centroid of the area from the y-axis.
1. Find the volume generated by revolving about the line y = 0 the area under the arch of the sine curve y = sin x from x = 0 to x = π 2. Find the volume of the solid generated by revolving about y =1 the area bounded by the curve y = e^x , OX and the lines x = –1 and x = 0. 3. Find the volume of the solids of revolution generated when the region bounded by the curve y = x^2 , the x – axis and the lines x = 1 and x = 2 is revolved about the x – axis.
The volume of a nose cone is generated by rotating the function y = x – 0.2x2 about the x-axis. What is the volume, in m3, of the cone. The volume of a nose cone is generated by rotating the function y = x – 0.2x2 about the x-axis. What is the volume, in m3, of the cone? What is the x coordinate of the centroid of the volume?
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. x = 1 − y2, x = y2 − 1 The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph. The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the second curve, changes direction at the point (0, 0.5), goes down and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the second curve, and exits the window in the fourth quadrant. The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the first curve, changes direction at the point (0, −0.5), goes up and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the first curve, and exits the window…
8. A water tank has a bowl shape formed by rotating the region bounded by y = x^2, x = 0 and y = 4 about the y axis. Therefore the height of the tank is 4 m and the radius at the top of the tank is 2 m. The tank is full of water. Find the amount of work needed to pump all of the water up out of the tank.
StokesTheorem.Evaluate∫ F·dr,whereF=arctanx/yi+ln√x2+y2j+k and C is the boundary of the triangle with vertices (0, 0, 0), (1, 1, 1), and (0, 0, 2).
Maximum surface integral Let S be the paraboloidz = a(1 - x2 - y2), for z ≥ 0, where a > 0 is a real number.Let F = ⟨x - y, y + z, z - x⟩. For what value(s) of a (if any)does ∫∫S(∇ x F) ⋅ n dS have its maximum value?
Plane Areas by Integration
Finding a center of mass Find the center of mass of a thin plateof density d = 3 bounded by the lines x = 0, y = x, and the parabolay = 2 - x2 in the first quadrant.
Consider the region bounded by the graphs of y = sin-1 x, y = 0, and x = 1/2. a. Find the area of the region. b. Find the centroid of the region.
*INTEGRAL CALCULUS Show complete solution (with graph). 6. Determine the centroid, C(x̅, y̅, z̅), of the solid formed in the first octant bounded by y = 4 − x^2 and x − z = 0.

SEE MORE QUESTIONS

Recommended textbooks for you

Calculus: Early Transcendentals

Calculus

ISBN:

9781285741550

Author:

James Stewart

Publisher:

Cengage Learning

Thomas' Calculus (14th Edition)

Calculus

ISBN:

9780134438986

Author:

Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher:

PEARSON

Calculus: Early Transcendentals (3rd Edition)

Calculus

ISBN:

9780134763644

Author:

William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher:

PEARSON

Calculus: Early Transcendentals

Calculus

ISBN:

9781285741550

Author:

James Stewart

Publisher:

Cengage Learning

Thomas' Calculus (14th Edition)

Calculus

ISBN:

9780134438986

Author:

Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher:

PEARSON

Calculus: Early Transcendentals (3rd Edition)

Calculus

ISBN:

9780134763644

Author:

William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher:

PEARSON

Calculus: Early Transcendentals

Calculus

ISBN:

9781319050740

Author:

Jon Rogawski, Colin Adams, Robert Franzosa

Publisher:

W. H. Freeman

Precalculus

Calculus

ISBN:

9780135189405

Author:

Michael Sullivan

Publisher:

PEARSON

Calculus: Early Transcendental Functions

Calculus

ISBN:

9781337552516

Author:

Ron Larson, Bruce H. Edwards

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS