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Calculus: Early Transcendental Functions
- Converting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/sqrt(x2 + y2) over the region 1<= x2 + y2<= e.arrow_forwardUSE COORDINATE CHANGE TO SLOVES THE DOUBLE INTEGRAL SHOWN IN THE PICTURE.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
- Volumes of solids Use a triple integral to find the volume of thefollowing solid. The solid bounded by the surfaces z = ey and z = 1 over the rectangle{(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ ln 2}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. dz dx dyarrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The wedge of the cylinder x2 + 4z2 = 4 created by the planesy = 3 - x and y = x - 3arrow_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_forwardEvaluating a Surface Integral. Evaluate ∫∫ f(x, y, z)dS, where S f(x,y,z)=√(x2+y2+z2), S:x2+y2 =9, 0⩽x⩽3, 0⩽y⩽3, 0⩽z⩽9.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_forward
- Integration by parts Evaluate the following integrals using integration by parts. ∫t2 e-t dtarrow_forwardFill in the blanks: A region R is revolved about the x-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_forwardLine integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful. The flux line integral of F = ⟨ex - y, ey - x⟩, where C is theboundary of {(x, y): 0 ≤ y ≤ x, 0 ≤ x ≤ 1}arrow_forward
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