Sketch the region bounded by the curves y=e^x ,y=-x+2 and x=0, generate an integral of type-I and type-II, representing the volume of the enclosed region.
Sketch the region bounded by the curves y=e^x ,y=-x+2 and x=0, generate an integral of type-I...
6. (a) (1 marks) Sketch the region bounded by the curves y = sin x, y = x+1, x = 0 and x = - 27. (b) (3 marks) Use the method of cylindrical shells to set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region about the line x = 27. (c) (3 marks) Use the method of washers to set up, but do not evaluate, an integral for the...
The region Bounded by the curves y=x2 is revolved about the x-axis. Give an integral for the volume of the solid that is generated. The region bounded by the curves y = 3x and y = x' is revolved about the x-axis. Give an integral for the volume of the solid that is generated. va | ndx (Type an exact answer using a as needed.)
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y- axis. Sketch the region enclosed by the curves and compute its area as an integral along the e- or y-axis. (a) 1 = \y, r = 1 - \yl. (b) 1 = 2y, 2 + 1 = (y - 1)2 21 c) y = cos.r, y = cos 2.c, I=0,2 = 3
Use a triple integral to compute the volume of the region bounded by curves y = 2-2x, x = 0,, and y=0 in the xy plane and the surface defined above by z = x^2
5. Sketch the region enclosed by the curves y = (x – 2)2 and y = x then find its area using the appropriate definite integral.
(a) For the double integral pin2 (In 2)2-y I = ef+y* dx dy. i. Sketch the region of integration. ii. Show that I = (extu 2) (b) Using a triple integral, calculate the volume of the region in the first octant (x > 0, y > 0, z > 0), bounded by the two cylinders z2 + y2 = 4 and x² + y2 = 4.
Consider a solid whose base is the region bounded by the curves y = (−x^2) + 3 and y = 2x − 5, with cross-sections perpendicular to the y-axis that are squares. a) Sketch the base of this solid. b) Find a Riemann sum which approximates the volume of this solid. c) Write a definite integral that calculates this volume precisely. (Do not need to calculate the integral)
4. Sketch the region enclosed by the curves y = x, y = 4x, y = -x +2, and find its area by any method. 5. Find the volume of the solid generated when the region between the graphs of f(x) = 1 + x2 and g(x) = x over the interval (0, 2) is revolved about the x- axis.
[4] Sketch the region bounded above the curve of y = x2 - 6, below y = x, and above y = -x. Then express the region's area as on iterated double integral ans evaluate the integral. -4 -3 -2 -1 0 1 2 3 4 [5] Find the area of the region bounded by the given curves x - 2y + 7 = 0 and y2 -6y - x = 0.
3. Sketch the region enclosed by the given curves and use a definite integral to calculate its exact area. y = 0,x=-1, y = 772 , x = 1