Calculus II

Consider the solid obtained by rotating the region bounded by the given curves about the y-axis.
y = ln x, y = 4, y = 5, x = 0
Find the volume V of this solid.

Help!!! Thank you in advance :(

  1. 👍 0
  2. 👎 0
  3. 👁 413
  1. This problem can be easily solved using the disk method.

    Horizontal disks are used, with slices of thickness dy.

    We will integrate from y=4 to y=5.

    Each disk has a volume of πr(y)²dy.
    where the radius is a function of y.

    Since y=ln(x), its inverse relation is x=e^y.

    Integrate for y=4 to 5 of
    V=∫π(e^y)²dy
    =π∫e^(2y)dy
    =π(1/2)e^(2y)
    Evaluate between 4 and 5 gives
    V=(π/2)(e^(2*5)-e^(2*4))
    =29917 (approx.)

    Check:
    The average radius is between e^4 and e^5=101.5
    Volume = 32400 approx. > 29917
    Since the curve ln(x) is concave up, the actual volume should be a little less than the approximation. So the calculated volume should be correct.

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 8 sin x, y = 8 cos x, 0 ≤ x ≤ π/4; about y = −1

  2. calculus

    1. Find the volume V obtained by rotating the region bounded by the curves about the given axis. y = sin(x), y = 0, π/2 ≤ x ≤ π; about the x−axis 2. Find the volume V obtained by rotating the region bounded by the curves

  3. calculus review please help!

    1) Find the area of the region bounded by the curves y=arcsin (x/4), y = 0, and x = 4 obtained by integrating with respect to y. Your work must include the definite integral and the antiderivative. 2)Set up, but do not evaluate,

  4. calculus

    1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln(5x), y = 1, y = 3, x = 0; about the y-axis 2. Use the method of cylindrical shells to find the volume V

  1. Calculus

    This problem set is ridiculously hard. I know how to find the volume of a solid (integrate using the limits of integration), but these questions seem more advanced than usual. Please help and thanks in advance! 1. Find the volume

  2. Calculus

    a) Find the volume formed by rotating the region enclosed by x = 6y and y^3 = x with y greater than, equal to 0 about the y-axis. b) Find the volume of the solid obtained by rotating the region bounded by y = 4x^2, x = 1, and y =

  3. calculus

    Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x+y=4,x=5−(y−1) 2 ;

  4. calculus

    Consider the solid obtained by rotating the region bounded by the given curves about the line x = 6. y = sqrt(x), y = x Find the volume V of this solid.

  1. calculus

    Sketch the region bounded by the curves y = x^2, y = x^4. 1) Find the area of the region enclosed by the two curves; 2) Find the volume of the solid obtained by rotating the above region about the x-axis; 3) Find the volume of the

  2. calculus

    Consider the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 9 - 9x^2 , y = 0 Find the volume V of this solid. Sketch the region, the solid, and a typical disk or washer. Any help or tips

  3. calculus

    Consider the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 6 x^6 , y = 6 x , x >= 0 Find the volume V of this solid.

  4. Calculus

    Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=x^2, x=y^2 about the axis x=–1

You can view more similar questions or ask a new question.