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Calculus
- Volumes of solids Use a triple integral to find the volume of thefollowing solid. The prism in the first octant bounded by z = 2 - 4x and y = 8.arrow_forward34) The figure shows the region of integration for the integral. Rewrite this itegral as an equivalent iterated integral in the five other orders. ∫0 to 1 ∫0 to (1-x^2) ∫0 to (1-x) f(x, y, z) dydzdxarrow_forwardComputing areas Use a double integral to find the area of thefollowing region. The region bounded by the spiral r = 2θ, for 0 ≤ θ ≤ π, and the x-axisarrow_forward
- USE COORDINATE CHANGE TO SLOVES THE DOUBLE INTEGRAL SHOWN IN THE PICTURE.arrow_forwardComputing areas Use a double integral to find the area of thefollowing region. The region bounded by the cardioid r = 2(1 - sin θ)arrow_forwardArea of regions Use a line integral on the boundary to find thearea of the following region. {(x, y): x2 + y2 ≤ 16}arrow_forward
- Integration by Substitution the integralsarrow_forward*INTEGRAL CALCULUS Show complete solution (with graph). 2. Determine the centroid of the area bounded by x^2 − y = 0 and x − y = 0.3. Determine the centroid of the area bounded by 2(y^2 + 4) − 2x − 8 = 0 and 8y + x^2 = 0.arrow_forwardSix orderings Let D be the solid in the first octant bounded bythe planes y = 0, z = 0, and y = x, and the cylinder 4x2 + z2 = 4.Write the triple integral of ƒ(x, y, z) over D in the given order of integration. dy dz dxarrow_forward
- Evaluating an iterated integral Evaluate V = ∫10 A(x) dx, whereA(x) = ∫20 (6 - 2x - y) dy.arrow_forwardIntegration by parts Evaluate the following integrals using integration by parts. ∫t2 e-t dtarrow_forwardRegions of integration Sketch each region R and write an iterated integral of a continuous function ƒ over R. Use the order dy dx. R = {(x, y): 0 ≤ x ≤ 4, x2 ≤ y ≤ 8√x}arrow_forward
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