Finding the Area of a Region In Exercises 31-36, (a) use a graphing utility to graph the region bounded by the graphs of the functions, (b) find the area of the region analytically, and (c) use the
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- EXAMPLE 6 Find the area of the region enclosed by the curves y = 1/x, y = x, and y = x, using (a) x as the variable of integration and (b) y as the variable of integration. SOLUTION The region is graphed in Figure 14.arrow_forwardSketch the graphs of the function f and g f(x) = x2 + 2, g(x) = 1 Find the area of the region enclosed by these graphs and the vertical lines x= 1 and x= 3arrow_forwardFinding the Area of a Region In Exercises 69–72, find the area of the region. Use a graphing utility to verify your result. | x/x + I dx x x + 2 dx 69. 70. y y 16 80 12 60 8 40 4 20 ++x 8 2 4 2 4 6 2.arrow_forward
- Find the area of the region bounded by x = y³ - 4y² + 3y and the x = y²-y. (a) A square units 77 12 12 (b) A= square units (c) A= 71 6 square units (d) A= square unitsarrow_forwardChapter Seven: Applications of Integration (Sketch all graphs) ( 7) Find the area of the region bounded by y = x² + 2x +1 and y = 2x + 5arrow_forwardSketch the region bounded by the graphs of the equations (include representative rectangle) f (x) = 3, g(x) = x – 1,and find the area of the region. Vx -1arrow_forward
- F2 (b) The area of the shaded region is Next Part F3 -2.13 + F4 F5 0 F6 H X F7 3 Соarrow_forwardAy y =x5 (0, у) (х, у) (0, 0) A right triangle has one vertex on the graph of y = x°, x> 0, at (x, y), another at the origin, and the third on the positive y-axis at (0, y), as shown in the figure. Express the area A of the triangle as a function of x. ...... A(x) =arrow_forwardSketch the graphs of the functions f and g. f(x) = x2 + 6, g(x) = 1 Then, find the area of the region enclosed by these graphs at the vertical lines x=1 and x=3arrow_forward
- Find the area of the region. f(x) = x-3/x The x y-coordinate plane is given. There is 1 curve and a shaded region on the graph. The curve starts at x = 3 on the x-axis, goes up and right becoming less steep, and ends at the approximate point (5, 0.40). The region below the curve, above the x-axis, and between 3 and 5 on the x-axis is shaded.arrow_forwardSymmetry Principle Let R be the region under the graph of y = f (x) over the interval [−a, a], where f (x) ≥ 0. Assume that R is symmetric with respect to the y-axis. (a) Explain why y = f (x) is even—that is, why f (x) = f (−x). (b) Show that y = xf (x) is an odd function. (c) Use (b) to prove that My = 0. (d) Prove that the COM of R lies on the y-axis (a similar argument applies to symmetry with respect to the x-axis).arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Geometry For College Students, 7eGeometryISBN:9781337614085Author:Alexander, Daniel C.; Koeberlein, Geralyn M.Publisher:Cengage,