Cartesian to polar coordinates Sketch the given region of
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- Consider the region bounded by the graphs of y = ln x, y = ex, x = 1, and x = e. Set up the definite integral that solves for the volume of the solid when the region is revolved about:a. the x-axis b. the y-axis c. the line x = -1 d. the line y = -1arrow_forwardHow do I classify whether the region is Type I or II? How can I approach the set-up of the integral in the problem? #68. The region D bounded by y=0, x=-10+y, and x=10-y as given in the following figure.arrow_forwardThe region of integration R is given by the following inequalities.R : x^2 + y^2 ≤ 1, x + y ≥ 1Evaluate the integral below by switching to polar coordinates. ∫ ∫(x − y)dAarrow_forward
- triple integral of 3xy dV, where E lies under the plane z = 1 + x + y and above the region in the xy-plane bounded by the curves y = square root of x, y = 0, and x = 1arrow_forwardA solid S is generated by revolving the region between the x-axis and the curve y =√ sinx (0 ≤ x ≤ π) about the x-axis.(a) For x between 0 and π, the crosssectional area of S perpendicular to the xaxis at x is A(x) = _____.(b) An integral expression for the volume of S is _____ .(c) The value of the integral in part (b) is_____ .arrow_forwardy = ex , y = 0 , x = 0 , x= ln2 Draw the region bounded by the curves y = e^x , y = 0 , x = 0 , x= ln2 in the first quartile. Express the area of this region as a double integral. Solve the integral.arrow_forward
- 34) 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_forwardA thick spherical shell occupies the region between two spheres of radii a and 2a, both centred on the origin. The shell is made of a material with density p = A(x2 + y2) z2, where A is a constant. Hence, or otherwise, find the mass of the shell by evaluating a suitable volume integral.You may find the substitution u = cosθ useful.arrow_forwardEvaluate the given DOUBLE integral by changing to polar coordinates. The double integral is: sqrt (16-x2-y2) dA where R = (x, y) | x2 + y2 ≤ 16, x ≥ 0.arrow_forward
- 4.Find first derivates y=(lnx)^lnx 5.Find the area of the region bounded by the graphs of x=3−y² and x=1+y. 6.Finding the Area of a Surface of Revolution. Set up and evaluate the definite integral forthe area of the surface generated by revolving the curve aboutthe y-axis. y=1−x²/4, 0≤x≤2arrow_forwardWork through all integrals. Determine the volumes of the solids of revolution generated by revolving the given region about the given line. Do by the method indicated. - The region bounded by y = sin(x) , y = 0, on [0,pi], is revolved about the y = 1. Do by washers.arrow_forwardNo integrals Let F = ⟨2z, z, 2y + x⟩, and let S be the hemisphereof radius a with its base in the xy-plane and center at the origin.a. Evaluate ∫∫S (∇ x F) ⋅ n dS by computing ∇ x F and appealing to symmetry.b. Evaluate the line integral using Stokes’ Theorem to check part (a).arrow_forward
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