Integration Handout B Applications of Integration 1 Write an integral or sum of difference of integrals giving the area of the region bounded above by the graph of y x2 2 and below by the graph of y x You need not evaluate 2 Find the area in the first quadrant bounded by y arcsin x y 2 and x 0 Hint To get an exact answer it will be simplest to integrate with respect to y 3 The following definite integrals can be computed exactly without knowing the antiderivative of arctan x The point of this problem is to interpret the definite integral given as the area under a curve and then to either use the symmetry of arctan x to evaluate the definite integral or chop the area into horizontal strips to arrive at different definite integral that is easy to evaluate Z 2 a arctan x dx b Z 2 1 arctan x dx 0 4 Write an integral that gives the volume generated by revolving the region bounded by y x 2 and y 4 about a the y axis b the vertical line x 2 c the horizontal line y 4 d the horizontal line y 1 You need not evaluate these integrals 5 A parfait cup is formed by revolving the curve y x 3 0 x 2 about the y axis The parfait cup is filled to the brim with hot chocolate If you plan to drink exactly half the hot chocolate in the cup what height should the liquid be when you stop drinking Feeling Blue Looking Bluer Spraying a piece of pottery with cobalt will result in a blue color when the piece is fired The shade of blue is determined by the density of cobalt the greater the density of the cobalt application the darker the blue of the pot You can get gradations of blue by applying cobalt glaze with a spray gun and varying the density of the application Makoto and Wasma are professional potters at the Radcliffe Pottery Studio on Western Avenue and the next four problems are about glazing pieces of pottery shades of blue 6 Makoto has made a rectangular sushi platter from a slab of clay 14 inches by 6 inches He applies cobalt such that the density of the application increases with the distance from one of the long sides of the platter The density of cobalt glaze is given by x mg square inch where x is the distance in inches from one long side of the sushi platter a How can you approximate the amount of cobalt Makoto used b Give an expression in terms of x that gives the amount of cobalt used 7 Makoto decides to try a more symmetric glaze application on his next sushi platter The platter is again 14 inches by 6 inches This time the deepest blue is in a stripe along the long center line of the platter and the intensity of the blue fades with the distance from this central line The density of cobalt glaze is given by x mg square inch where x is the distance in inches from the longitudinal center of the sushi platter a How can you approximate the amount of cobalt Makoto used b Give an expression in terms of x that gives the amount of cobalt used 8 Wasma is glazing a large round plates 16 inches in diameter For one plate she decides to have a deep blue center fading out into pale blue along the rim She applies cobalt glaze such that its density is given by x mg square inch where x is the distance in inches from the center of the plate 1 a How can we approximate the amount of cobalt Wasma used b Write an expression in terms of x that gives the amount of cobalt used 9 For her next round plate 16 inches in diameter Wasma decides to have a deep blue line 16 inches long running through the center of the plate She has the shade of blue fade into paler and paler blue as the distance from this deep blue line increases She applies cobalt glaze such that its density is given by x mg square inch where x is the distance in inches from dark blue diameter of the plate Write an expression in terms of x that gives the amount of cobalt used 1010 10 The density of dart holes on an old dartboard is given by x x 2 1 holes per square inch where x is the distance in inches from the center of the board If the board is a circle with diameter 20 inches find the total number of holes in the board 11 Given a disk of radius R suppose you partition the interval 0 R into n equal pieces each of length x R n Typically in our work we have let xk k x so x0 0 x1 x x2 2 x and xn n x R and then approximated the area of the kth ring by 2 x x There are several ways to justify the validity of using that approximation in our work This problem asks you to work through one of them Let x k be the midpoint of the kth interval Then the left and right hand endpoints of the kth interval can be written as x k 21 x and x k 12 x respectively Show that the area of the kth ring computed using x k 21 x as the inner radius and x k 21 x as the outer radius is exactly 2 x k x Conclude that the approximation we have been using is a valid one 12 The density of a ball of ice is greatest at the center and decreases with the distance from the center of the ball The ball is 10 centimeters in radius and the density is given by x grams per cubic centimeter What is the mass of the ball 13 A chocolate truffle is a wonderfully decadent chocolate concoction Truffles tend to be spherical or hemispherical a Consider a truffle made by dipping a round hazelnut into various chocolates building up a delicious chocolate delicacy The number of calories per cubic millimeter varies with x the distance from the center of the hazelnut If x gives the calories per cubic millimeter at a distance x millimeters from the center write an integral that gives the number of calories in a truffle of radius R b Another truffle is made in a hemispherical mold of radius R The mold looks like a tiny hemispherical bowl Different layers of chocolate are poured into the mold one at a time and allowed to set The number of calories per cubic millimeter varies with x the distance from the top of the mold The caloric density is given by x calories per cubic millimeter Write an integral that gives the number of calories in this hemispherical truffle 14 In the town of Lybonrehc there has been a nuclear reactor meltdown which released radioactive iodine 131 Fortunately the reactor has a containment building which kept the iodine …