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Review Fundamental Concepts and Techniques of Calculus Exercises:∗ Integral Calculus Last updated: 041012 1. Integration By Parts: 1 (s) sin−1 (ln x2 ) dx S x (a) xex dx HSD √ (t) sin x dx S (b) x3 3x dx S 2. Trigonometric Integrals: (c) x sin x dx HSD (a) tan3 x sec3 x dx HSD (d) ex sin x dx HSD (b) sin4 (3t) cos(3t) dt S (e) ln x dx S (c) sin4 x cos3 x dx HSD (f) xn ln x dx S (d) sec3 (6x) dx S (g) x cos πx dx S (e) cos6 x dx HSD (h) cos(ln x) dx S (f) sin4 x cos4 x dx HSD (i) x tan−1 (x2 ) dx S (g) (cos 3x − sin x)2 dx S 1 (j) sin−1 (ln x) dx S x (h) tan4 x dx S √ (k) cos x dx S 3 (i) tan x sec 2 x dx S (l) x ln(x + 1) dx S (j) cos3 x cos2 x dx S −1 sin 2x (m) √ dx S 1 − 4x2 (k) csc3 1 2x dx S (n) x2 cosh 2x dx S (l) sin 5x cos 2x dx S (o) x2 (2x − 1)−7 dx S (m) sec5 x dx S (p) x tan−1 x2 dx S (n) cot2 x csc2 x dx S (q) sin−1 2x dx S 3. Substitution: √ (r) x2 tan−1 (x2 ) dx S (a) 1+ x dx HSD ∗ The letters “H”,”S”, “D” in the status box indicate whether a hint (H), a solution (S) or a detailed solution (D) is available. 1 3 (b) t5 1 + t2 dt S (z) x2 e−x dx HS (c) x2 (ex − 1) dx S 4. Partial Fractions: 7 sin t (a) dx S (d) dt S (x − 2)(x + 5) 1 + cos t √ x 1+ x+1 (b) dx S (e) √ dx S (x + 1)(x + 2)(x + 3) x+1 dx tan(ln t) (c) S (f) dt S (x − 1)3 t x5 sin−1 2x (d) dx S (g) √ dx HSD (x − 2)2 1 − 4x2 x 2x2 + 3 (h) √ dx (e) dx S 2−4 x2 (x − 1) x x x2 + 3 (i) √ dx (f) dx S 4 − x2 x2 − 3x + 2 1 −3x (j) √ dx (g) 2 − x − 12 dx S x2 x2 − a2 x x x−3 (k) √ dx (h) dx S 6x − x2 x2 + 4x + 1 7 + 6x (l) (x2 + 3)(x3 + 9x + 1) 3 dx 1 S (i) dx S (6x + 1)2 √ e x x3 + x2 + x + 3 (m) √ dx S (j) dx S x (x2 + 1)(x2 + 3) x2 − 3x − 1 (n) x x2 + 6x dx (k) dx S x3 + x2 − 2x x4 − x3 − x − 1 (o) x tan−1 x dx (l) dx S x3 − x2 (p) x sin−1 x dx (m) e2x tan2 (e2x ) sec2 (e2x ) dx S ln(x + 1) 1 − cos x (q) √ dx S (n) dx S x+1 1 + sin x 1 x 1 3 25x2 + 21 + 2x (r) dx S (o) dx S x2 x+1 (x + 3)(5x2 + 3) ex dx (s) √ dx HSD (p) 9 − e2x [(2x + 1)2 + 9]2 x+3 dx (t) √ dx HSD (q) x2 + 4x + 13 (x2 + 2x + 2)2 x dx dx (u) √ S (r) 4 − x2 x4 − 16 dx (v) 6x − x2 − 8 dx S (s) √ 1− x x dx (w) √ dx S (t) x−1 x(x1/3 − 1) √ tan2 x x (x) dx S (u) √ dx x − tan x x−1 dx x+1 (y) √ dx HSD (v) √ dx x2 a2 + x2 x x−2 2 1 √ (w) dx iii. {(x, y) | 3 ≤ x ≤ 5, x ≤ y ≤ x2 }, 1 + cos x − sin x rotated about the line x = 5 1 √ (x) dx iv. {(x, y) | 3 ≤ x ≤ 5, x ≤ y ≤ x2 }, 2 + cos x rotated about the line y = 1 1 v. Find the volume generated by revolving an (y) dx sin x + tan x isosceles right triangle with leg a about a line perpendicular to the hypotenuse and 5. Improper Integrals: through a vertex (not the right-angle ver- ∞ tex). dx (a) 1 x2 (d) Arc Length: ∞ dx i. C : y = x2 , 0 ≤ x ≤ 2 (b) 1 4 + x2 8 ii. C : y = ln cos x , 0 ≤ x ≤ π/4 dx (c) (e) Surface Area: 0 x2/3 1 i. C := {(x, y) | x2 +y 2 = 1, x ≥ 0, y ≥ 0} dx (d) √ revolved about the line y = 1 0 1 − x2 ∞ ii. C := {(x, y) | x2 +y 2 = 1, x ≥ 0, y ≥ 0} ln x revolved about the line x = 2 (e) dx e x 3 iii. C : y = cos x, 0 ≤ x ≤ π/2 dx (f) √ 3 iv. A square with side s is revolved about a line 1/3 3x − 1 parallel to a diagonal and a distance r from 6. Applications of Integration: the nearest vertex. (f) Mass From Density: (a) Area Between Two Curves: i. Find the mass of a rod of length whose i. y = 2x, y = 4 − x, and y = −x/2 density varies as the square of the distance ii. Under |x2 − 1| over [−3, 1] from one end. iii. Between y = cos x and y = sin x over ii. Find the mass of a semi-circular disc of ra- [0, π/4] dius a in which density varies as the square (b) Volume By Slicing: of the distance from the diameter. √ iii. Find the mass of the solid circular cylin- i. Under y = 1 − x2 , rotated about the line y = −1 der of radius r and length whose density √ varies as the distance from the axis of sym- ii. Under y = 1 − x2 , rotated about the line metry. x=1 iv. Find the mass of a circular disc of radius 4 iii. The base of the solid is the circular region in which density varies as the distance from x2 + y 2 ≤ 1. A cross-section perpendicular the center. to the y-axis is a square. iv. The base of the solid is the semi-circular (g) Center of Mass and Centroid: √ region under y = 1 − x2 over [−1, 1]. A i. Find the center of mass of a rod of length cross-section perpendicular to the y-axis is 2 whose density varies as the distance from an equilateral triangle. one end. v. Two cylinders of radius r intersect at right ii. Find the center of mass of a plate covering angles, their axes meeting. Find the volume the region below y = x2 over [0, 2] whose of the intersection. What is the shape of a density var ies as the distance from the line cross-section parallel to the plane of the two x = 3. axis? iii. Find the center of mass of a wire covering (c) Volume By Shells: the unit circle whose density varies as the i. {(x, y) | 0 ≤ x ≤ 2, 0 ≤ y ≤ x3 }, rotated distance from the line x = 2. about the line x = 3 iv. Find the centroid of a semi-circular disc of √ radius a ii. {(x, y) | 3 ≤ x ≤ 5, x ≤ y ≤ x2 }, rotated about the line x = 1 v. Find the centroid of a solid cone. 3 vi. Find the centroid of a solid parabolic cone iv. x(t) = e6t , y(t) = e3t − 3 obtained by revolving the region below y = √ v. x(t) = 2 sin t, y(t) = cos t x over [0, a] about the x-axis. (b) Find a Parametrization x = (t), y = y(t), vii. Find the centroid of a hemispherical bowl t ∈ [0, 1] for of radius a. viii. Find the centroid of the arc y = ln x, 1 ≤ i. The line segment from (3, 7) to (8, 5). x ≤ e. ix. Use Pappus’ Theorem to ﬁnd the volume of ii. The line segment from (2, 6) to 6, 3). the solid generated by revolving the region iii. The parabolic arc x = 1 − y 2 from (0, −1) below y = sin x over [0, π] about the line to (0, 1). x = 2π. iv. The curve y 2 = x3 from (4, 8) to (1, 1). (h) Fluid Pressure: i. A water tank is a cylinder 6 feet in diameter, (c) Find a parametrization x = (t), y = y(t) for the standing on end. If water ﬁlls the tank to a hyperbola x2 − y 2 = 1. depth of 4 feet, how much force is exerted (d) Find the slope of the given curve at the given on the sides of the tank? point and give an equation of the tangent line: ii. The face of a dam is an isosceles trapezoid i. x(t) = 2 cos t, y(t) = 3 sin t, t = π/3 of height 20 feet and bases 10 feet and 40 feet. It is situated vertically with the smaller base at the bottom. Find the force on the ii. x(t) = 1 + t, y(t) = 1 − t2 , t = 2 dam if the water level is 2 feet below the (e) Find the Length of the Parametric Arc: top of the dam. i. x(t) = t, y(t) = t2 , t ∈ [0, 3] (i) Work: ii. x(t) = 2 sin t, y(t) = 2 cos t, t ∈ [0, π/2] i. Find the work done in moving on object from x = 1 to x = 5 if the force acting √ at point x is given by F (x) = x2 + 1. 8. Polar Coordinates: (a) Write the Equation in Polar Coordinates: ii. A vertical cylindrical tank of radius 2 feet and height 6 feet is full of water. Find the i. x = 2 work done in pumping out the water (a) to ii. x2 + (y − 1)2 = 4 an outlet at the top of the tank; (b) to a level iii. y = mx 5 feet above the top of the tank. (Assume that the water weights 62.5 pounds per cu- (b) Write the Equation in Cartesian Coordinates: bic foot.) i. r sin θ = 4 iii. A conical container (vertex down) of radius ii. tan θ = 2 r feet and height h feet is full of a liquid iii. r = 4 sin(θ + π) weighing σ pounds per cubic foot. Find the 1 (c) Sketch the Polar Curves: work done in pumping out the top 2 h feet of liquid: (a) to the op of the tank; (b) to a 1 i. θ = − 4 π level k feet about the top of the tank. ii. r = −2 sin θ iv. A chain that weights 15 pounds per foot is hanging from the top of an 80-foot building iii. r = 1 − cos θ to the ground. How much work is done in iv. r = cos 2θ pulling the chain to the top of the building? v. r = sin 3θ vi. r = −1 + 2 cos θ 7. Parametric Curves: (d) Calculate the Area enclosed by the Polar Curve: (a) Express the Parametric Curve by an Equation in i. r = a cos θ, θ ∈ [− 1 π, 1 π] 2 2 x and y: 1 ii. r = a(1 + cos 3θ), θ ∈ [− 1 π, 3 π] 3 i. x(t) = 3t − 1, y(t) = 5 − 2t iii. Outside r = 2, but inside r = 4 sin θ ii. x(t) = 2t − 1, y(t) = 8t3 − 5 iv. Inside r = 4, and between the lines θ = 1 π 2 iii. x(t) = e2t , y(t) = e2t − 1 and r = 2 sec θ 4 v. Inside the inner loop of r = 1 − 2 sin θ iii. r = 1 + cos θ; where are the tangent lines either horizontal or vertical? (e) Find the Slope of the Polar Curve: (f) Find the Length of the Polar Curve: i. r = a, [a, π/3] i. r = 3 cos θ √ ii. r = a(1 + cos θ) ii. r = sin 2θ, [ 3/2, π/6] 5