Applications of Integration

1. 1. Evaluate the integral (leave your answer in exact form): TT / 4 3 sin(2.x) esin'(x) d.x TT / 6 2. 2. Evaluate the integral: In(esec= (7x) )dx111 Exercises find the average value of the function over the indicated interval. Find a point in this interval at which the function takes on the average value. f(x)=6x2., xe [0,2] 4. 4. The Gateway Arch in St. Louis is [53011 high and has a 'dU-ft base. 11.5 shape can be modeled by the parabola y = r330[1_ (TH Find the average height 01' the arch above I.h(.' ground. 5. 5. Use the open space to show all of your work. This includes: sketching the curves, finding and labeling the points of intersection, setting up the appropriate definite integral(s) and evaluating them. Place your numerical answer for the area of the region in the box. Leave your answer in exact form. Sketch the region bounded by the curves and find the area of that region. y= x, y= 5x-x3 6. 6. Find the area between the 2 curves by integrating with respect to y. Use the open space to show all of your work. This includes: sketching the curves, finding and labeling the points of intersection, setting up the appropriate definite integral(s) and evaluating them. Place your numerical answer for the area of the region in the box. Leave your answer in exact form. Sketch the region bounded by the curves and find the area of that region. x=y', x - 3y =27. The base of a solid is the circle x2 + y =16. Find the volume of the solid given that the cross sections perpendicular to the y - axis are squares. 8. 8. Sketch the region bounded by the following curves: y = 2x", y = 16, x = 0. Find the volume of the solid generate by revolving that bounded region about the y-axis.9. Let H. be the region bounded by the following curves: 3; = {75 :1: = 8, y = 0. [a] Sketch the region and label the vertiees of B. Set up the integration to nd the volume of the solid generated by revolving 1?. around: [\\DM 10. Find the an: length of the following curve by integrating with respect to at: 3; (ex + (:71) en the interval [7111(2),111(2)]