FROM ROGAWSKI S CALCULUS (2ND ED.) SECTION 5.4 18.) Express the antiderivative F (x) of f(x) satisfying the given initial condition as an integral. f(x) = x + 1 x 2 + 9, F (7) = 28.) Find G (1), where G(x) = x 2 t3 + 3 dt. 32.) Calculate d x 4 dx x t dt. 2 39.) Let A(x) = x f(t) dt, with f(x) as in Figure 11. (a) Where does A(x) have a local maximum at P? (b) Where does A(x) have a local minimum? (c) Where does A(x) have a local maximum? (d) True or false? A(x) < for all x in the interval shown. 42.) Find a b such that b a (x2 9) dx has minimal value. 44.) Let A(x) = x f(t) dt, and match the property of A(x) with the corresponding property of the graph of f(x). Assume f(x) is a differentiable.
FROM ROGAWSKI S CALCULUS (2ND ED.) 48.) Figure 13 shows the graph of f(x) = x sin x. Let F (x) = x t sin t dt. (a) Locate the local max and absolute max of F (x) on [, 3π]. (b) Justify graphically: F (x) has precicely one zero in [π, 2π]. (c) How many zeros does F (x) have in [, 3π]? (d) Find the inflection points of F (x) on [, 3π]. For each one, state whether the concavity changes from up to down or from down to up. SECTION 5.5 1.) A particle moves in a straight line with the given velocity. Find the displacement and distance traveled over the time interval, and draw a motion diagram like Figure 3 (see chapter
for example) (with distance and time labels). v(t) = 36 24t + 3t 2, [, 1] 13.) Find the net change in velocity over [1, 4] of an object with a(t) = 8t t 2 m/s 2. 14.) Show that if acceleration is constant, then the change in velocity is proportional to the length of the time interval. 2.) To model the effects of a carbon tax on CO 2 emissions, policy-makers study the marginal cost of abatement B(x), defined as the cost of increasing CO 2 reduction from x to x + 1 tons (in units of ten thousand tons - Figure 4). Which quantity is represented by the area under the curve over [, 3] in Figure 4? 24.) The heat capacity C(T ) of a substance is the amount of energy in joules required to raise the temperature of 1 g by 1 C at temperature T. (a) Explain why the energy required to raise the temperature from T 1 to T 2 is the area under the graph of C(T ) over [T 1, T 2 ]. (b) How much energy is required to raise the temperature from 5 to 1 C if C(T ) = 6 +.2 T? 1.) Show that the function is concave up on the whole real line. AND THE FOLLOWING PROBLEMS: f(x) = x (sin 4 t + 5t) dt 2.) A particle moves along a straight line. Its displacement at time t from an initial point on the line is given by the formula f(t) = 1 2 t2 + 2t sin t for t π. From t = π and afterwards the particle moves with constant velocity, i.e. the velocity it acquires at time t = π. Compute the following:
(a) Find its velocity at t = π (b) Find its acceleration at t = 1 2 π. (c) Find its displacement from at time 5 2 π FROM ROGAWSKI S CALCULUS (2ND ED.) (d) Find a time t > π when the particle returns to the initial point, or else prove that it never returns to. PART II SECTION 5.6 72.) Use the change of variables formula to evaluate the definite integral. 2 79.) Evaluate 2 r 5 4 r 2 dr. 1 4x + 12 (x 2 + 6x + 1) 2 dx SECTION 6.1 15.) Find the area of the shaded region in the figure. 22.) Figure 18 shows the region enclosed by y = x 3 6x and y = 8 3x 2. Match the equations with the curves and compute the area of the region.
In 29 and 38, sketch the region enclosed by the curves and compute its area as an integral along the x- or y- axis. 29.) x + y = 4, x y =, y + 3x = 4 38.) x + y = 1, x 1/2 + y 1/2 = 1 48.) Express the area (not signed) of the shaded region in Figure 2 as a sum of three integrals involving f(x) and g(x). SECTION 6.2 6.) Find the volume of the wedge in Figure 2(A) by integrating the area of vertical cross sections. 8.) Let B be the solid whose base is the unit circle x 2 + y 2 = 1 and whose vertical cross sections perpendicular to the x-axis are equilateral triangles. Show that the vertical cross sections have area A(x) = 3(1 x 2 ) and compute the volume of B. 14.) Find the volume of the solid with the given base and cross sections. The base is the region enclosed by y = x 2 and y = 3. The cross sections perpendicular to the y-axis are rectangles of height y 3. 2.) A plane inclined at an angle of 45 passes through a diameter of the base of a cylinder of radius r. Find the volume of the region within the cylinder and below the plane. 22.) Let S be the intersection of two cylinders of radius r whose axes intersect at an angle θ. Find the volume of S as a function of r and θ.
FROM ROGAWSKI S CALCULUS (2ND ED.) SECTION 6.3 8.) Find the volume of revolution about the x-axis for the given function and interval. f(x) = 4 x 2, [, 2] In 16 and 18, (a) sketch the region enclosed by the curves, (b) describe the cross section perpendicular to the x-axis located at x, and (c) find the volume of the solid obtained by rotating the region about the x-axis. 16.) y = x 2, y = 2x + 3 18.) y = 1 x, y = 5 2 x 26.) Let R be the region enclosed by y = x 2 + 2, y = (x 2) 2 and the axes x = and y =. Compute the volume V obtained by rotating R about the x-axis. Hint: Express V as a sum of two integrals. 3.) Find the volume of the solid obtained by rotating region A in Figure 13 about the y-axis. 46.) Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis. y = 2 x, y = x about y = 4 56.) The torus (doughnut-shaped solid) in Figure 15 is obtained by rotating the circle (x a) 2 + y 2 = b 2 around the y-axis (assume that a > b). Show that it has volume 2π 2 ab 2. Hint: Evaluate the integral by interpreting it as the area of a circle.
AND THE FOLLOWING PROBLEMS: 1.) Find c such that the area between the curves x 4 c and c x 4 is 1. 2.) Compute the volume of the solid formed by rotating the region bounded by y = 4 x 2, y = 2 + x, x = 2, and the y-axis around the line y = 4.