Surface area with double integrals

Surface area with double integrals

Lessons

Notes:

Surface Area with Double Integrals

Suppose we want to find the surface area given by the function f(x,y)f(x,y) from the region DD. Then the surface area can be calculated using the following:

S=D[fx]2+[fy]2+1dAS = \int \int_D \sqrt{[f_x]^2 + [f_y]^2 + 1} dA

  • Introduction
    Surface Area with Double Integrals Overview:
    a)
    • Surface Area with a function with Region DD
    • Partial Derivatives

    b)
    • Find the Region DD
    • Find the partial derivatives fxf_x & fyf_y
    • Calculate the Double Integral


  • 1.
    Finding the Surface Area with Double Integrals
    Determine the surface area of the surface x+2y+2z=4\, x + 2y + 2z = 4 that is in the 1st octant.

  • 2.
    Determine the surface area of the surface z=2x2y2 \, z = 2 - x^{2} - y^{2} that is above z=1+x2+y2 z = 1 + x^{2} + y^{2} with x0 x \geq 0 and y0 y \geq 0 .

  • 3.
    Determine the surface area of the surface y=3x2+3z24 \, y = 3x^{2} + 3z^{2} - 4 that is inside the cylinder x2+z2=1 x^{2} + z^{2} =1.