Concept explainers
Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [α, β], which implies that the area bounded by the graph of h and x-axis on [α, β] equals
104. Show that the area of the region bounded by the ellipse x = 3 cos t, y = 4 sin t, for 0 ≤ t ≤ 2π, equals
Trending nowThis is a popular solution!
Chapter 12 Solutions
Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
Thomas' Calculus: Early Transcendentals (14th Edition)
Precalculus Enhanced with Graphing Utilities (7th Edition)
University Calculus: Early Transcendentals (3rd Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
University Calculus: Early Transcendentals (4th Edition)
- Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.arrow_forwardhi My question is about Complex Derivative. I showed in the upload photo. z is a complex number and z* is complex conjugate of z. f(z)=u(x,y)+i.v(x,y); In section a, we want to obtain the derivative of the function f with respect to z and z * in the Cartesian coordinate system. In section b, I want to obtain the derivative of the function f with respect to z and z * in the polar coordinate system. . In one of the photos, I put a pattern to solve the question. Thank you very much.arrow_forwardIn Basic Calculus. Thank youarrow_forward
- Calculus Distance between point P (2, -1, -3) and line 11 with equation given by (parametrically): { x(t) = 1 - 3t; y(t) = 4 - 2t; z(t) = 3/2 - t/2}arrow_forwardThe curve in the plane defined by the equation (*) (x^2 + xy + y^2 = 7) is a rotated ellipse, shown in the figure. (c) The curve has two horizontal tangent lines. Find their equations.(d) Why can we see from the figure that the curve is not the graph of a function y of x?Show directly from (∗) that y is not a function of x.arrow_forwardplease help!arrow_forward
- ses/ 49385_1/cl/outline uestion Completion Status: O a. V() - 18,000 Ob.V) - 108,000-18,000r O -18,000r-108,000 Od.)-18,000r + 108,000 O e.V(t) - 108,000 QUESTION 8 Detemine whether the equation defines y as a linear function of x. If so, write it in the for y=mx + 6. + 3y= 0 Oby=-x O e.y is not a linear function of x QUESTION 9 Determine whether the equaticn defines y as a linear function of x. If so, 117ite it in the form y= mx-b. U 0: a linear funouon ofaarrow_forwardFind the curve y if dy = x2 +5, dx +5, curve passes through (0, 25)***arrow_forward5. C nts) Let F =, use Green's Theorem to evaluate F. dr where the curve C' is the curve shown in the picture consisting of functions y=x² − 2x going from point (0,0) to the point (3,3) followed by the line segment from (3,3) to (0,0). (0,0) y=x (3,3) C y=x²-2xarrow_forward
- Considering the following graph of the given function f. у The xy-coordinate plane is given. The curve enters the window in the second quadrant nearly horizontal, goes down and right, crosses the negative x-axis, changes direction on the negative y-axis, goes up and right, crosses the positive x-axis, and exits the window in the first quadrant nearly horizontal. Use the graph of f to complete the following table. f(x) f'(x) x > 0 f(x) > 0 f'(x) ? v 0 x > 0 f(x) 0 f'(x) ? v 0 x < 0 f(x) < 0 f'(x) ? v 0 Sketch the graph of f and f' on the same coordinate axes. y y yarrow_forwardFind parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=4et,y=te2t,z=tet^2;(4,0,0) Solve x(t), y(t), z(t)arrow_forward17. Sketch the ioiicwing curve by cing seconá derivative: 1) y= 1+x 2) y=-x(x-7) 3) y (x+ 2) (x-3) 4) y=x(5-x) (ans.: max.(1,0.5); min.(-1,-0.5)) (ans.: max.(7,0); min.(2.3,-50.8)) (ans.: max.(-2,0); min.(1.3,-18.5)) (uns. mux.(3.5,18.5);ra0,0)) 18. What is the smallest perimeter possible for a rectangle of area 16 in.2 ? (ans.: 16) 19. Find the area of the largest rectangle with lower base on the x- axis and upper vertices on the parabola y 12-x. (ans.:32) 20) A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. With 800 m. of fence at your disposal. What is the largest area you can enclose ? (ans.:80000) 21) Show that the rectangle that has maximum area for a given perimeter is a square. 22) A wire of length L is available for makıng a circie and a square. How should the wire be divided between the two shapes to maximize the sum of the enclosed areas? (ans.: all bent into a circle) 23) A closed container is made from a…arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage