CONCEPT CHECK
Choosing a
In Exercise 1 and 2, the region
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Chapter 14 Solutions
Multivariable Calculus
- Sine football Find the volume of the solid generated when the region bounded by y = sin x and the x-axis on the interval [0, π] is revolved about the x-axis.arrow_forwardUsing polar coordinates, evaluate the integral (sin(x2+y2)dA) over the region 1<=x2+y2<=81.arrow_forwardSurface areas Use a surface integral to find the area of the following surfaces. The surface ƒ(x, y) = √2 xy above the polar region{(r, θ): 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}arrow_forward
- Converting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/sqrt(x2 + y2) over the region 1<= x2 + y2<= e.arrow_forwardConverting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/(x2 + y2) over the region 1<= x2 + y2<= e^2.arrow_forwarda) Sketch planar region b)Sketch the solid c)Draw and label a cross section. Use the cross-section method to SET UP an integral for ? d)Draw and label a cylindrical shell. Use the shell method to SET UP an integral for V e) Evaluate one intergral in c) and d) respectivelyarrow_forward
- Cartesian to polar coordinates Evaluate the following integralover the specified region. Assume (r, θ) are polar coordinates.arrow_forwardLine integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful. The circulation line integral of F = ⟨x2 + y2, 4x + y3⟩, where Cis the boundary of {(x, y): 0 ≤ y … sin x, 0 ≤ x ≤ π}arrow_forwardSet up the integral (do not evaluate) in (u, v) coordinates to find the integralSSRx + y dA,where R is the parallelogram with vertices (0, 0), (1, −1), (5, 0), (4, 1). Make the substitutionx = 4s + t, y = s − t. You must draw the correspoding s, t region. The S' are the integral symbol since I cannot find copy paste versions of them on the internet.arrow_forward
- A). Use Pappus's theorem for surface area and the fact that the surface area of a sphere of radius d is 4pid^2 to find the centroid of the semicircle x=(d^2-y^2)^0.5arrow_forwardRegion B: Computing the integral of the function f (x, y) = (x + y) cos (x + y), with a triangle consisting of vertices (0,0), (a, a) and (a, -a).arrow_forwardFill in the blanks: A region R is revolved about the y-axis. The volume of the resulting solid could (in principle) be found by using the disk>washer method and integrating with respect to__________________ or using the shell method and integrating with respect to ___________________.arrow_forward
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