EXAMPLES4
DOUBLE INTEGRALS
1997 1. Evaluate the following integral by changing to polar coordinates:
1998 2. An annular washer of constant surface density δ occupies the region between the circles
x2 + y2 = a2 and x2 + y2 = b2 where b > a.
Find the moment of inertia of the washer about the x-axis.
2014, S1
2014, S1
2014, S2
2015, S1
2015, S2
3. Consider the double integral
i) Sketch the region of integration.
ii) Evaluate I using polar coordinates.
4. A thin triangular plate bounded by y = 2x, y = 6 and the y axis has non-uniform density given by ρ(x,y) = 4xy. Find the mass of the plate by evaluating an appropriate double integral in Cartesian coordinates.
5. Consider the double integral
i) Sketch the region of integration.ii) Evaluate the double integral by first converting to polar coordinates.
6. Consider the double integral
i) Sketch the region of integration.ii) Evaluate the double integral by first reversing the order of integration.
7. The area A of a region R of the xy-plane is given by
i) Sketch the region R.
ii) When the order of integration is reversed the expression for A becomes
Find the limits l1(x) and l2(x).
iii) Hence, find the value of A.
i) Sketch the region of integration. ii) Evaluate the double integral using polar coordinates.
,
where z is the coordinate measured along the axis of rotation, a = 6378 km is the radius of the Earth at the equator and b = 6357 km is the radius of the Earth at the poles.
Calculate the volume of the Earth using an appropriate double integral.
i) Sketch the plate in the x y plane.
−
ii) Without evaluating any integrals write down the mass of the plate.
iii) Find the coordinates of the centroid (¯x,y¯) of the plate by evaluating an appropriate double integral in polar coordinates. (Note that by symmetry, ¯y = ¯x).
i) Sketch the region of integration. ii) Evaluate I by first reversing the order of integration.
i) Sketch the region of integration.
− − the plane z = 0.
Determine the area of the region enclosed by the curve by using a suitable double integral.
i) Sketch the region of integration. ii) Evaluate I with the order of integration reversed.
uniform density and has centroid (¯x,y¯).
i) Sketch the region Ω and write down its area.
ii) Explain why ¯x = 0. iii) Find ¯y by evaluating an appropriate double integral expressed in polar coordinates.
