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Review Fundamental Concepts and Techniques of Calculus Hints to the Exercises: Integral Calculus Last updated: 041012 1. Integration By Parts: (c) Use the ﬁrst Pythagorean identity and rewrite (a) Use integration by parts (R 4:6c) and set u = x cos3 x = cos2 x cos x = (1 − sin2 x) cos x. and v = ex . Then apply (R 4:5b). (b) (d) (c) Use integration by parts (R 4:6c) and set u = x (e) You can use the trigonometric identity cos2 x = and v = sin x 1 2 (1 + cos 2x) to reduce the power of cos in (d) Integrate by parts (R 4:6c) twice such that the the integrand or apply the recursion formula remaining integral is a multiple of the original (R 4:5q). integral. Combine those integrals on the left side (f) Use the trigonometric identities of the equation. 1 (e) sin x cos x = 2 sin(2x) 2 1 (f) cos x = 2 1 + cos(2x) 2 1 (g) sin x = 2 1 − cos(2x) (h) to reduce the number of factors. (i) (g) (j) (h) (k) (i) (l) (j) (m) (k) (n) (l) (o) (m) (n) (p) (q) 3. Substitution: (r) (a) √ the substitution (R 4:6b) given by y = 1 + Try (s) x. (t) (b) (c) 2. Trigonometric Integrals: (d) (a) Use the Second Pythagorean identity and (e) rewrite (f) tan3 x sec3 x = sec2 x tan2 x(sec x tan x) (g) Try the substitution (R 4:6b) given by y = sin−1 (2x). = sec2 (sec2 x − 1)(sec x tan x). (h) Then apply (R 4:5b). (i) (b) (j) 1 (k) (w) (l) (x) (m) (y) (n) 5. Improper Integrals: (o) (a) (p) (b) (q) (c) (r) (d) (s) Reduce the integrand by the substitution (e) (R 4:6b) given by y = ex . (f) (t) Complete the square and use trigonometric sub- stitution (R 4:6(b)iii). 6. Applications of Integration: (u) (a) Area Between Two Curves: (v) i. (w) ii. (x) iii. (y) Use trigonometric substitution (R 4:6(b)iii) and (b) Volume By Slicing: set x = a tan θ. i. 3 (z) let u = −x ii. iii. 4. Partial Fractions: iv. (a) v. (b) (c) Volume By Shells: (c) i. (d) ii. (e) iii. iv. (f) v. (g) (d) Arc Length: (h) i. (i) ii. (j) (e) Surface Area: (k) i. (l) ii. (m) iii. (n) iv. (o) (f) Mass From Density: (p) i. ii. (q) iii. (r) iv. (s) (g) Center of Mass and Centroid: (t) i. (u) ii. (v) iii. 2 iv. (e) Find the Length of the Parametric Arc: v. i. vi. ii. vii. 8. Polar Coordinates: viii. ix. (a) Write the Equation in Polar Coordinates: (h) Fluid Pressure: i. i. ii. ii. iii. (i) Work: (b) Write the Equation in Cartesian Coordinates: i. i. ii. ii. iii. iii. iv. (c) Sketch the Polar Curves: 7. Parametric Curves: i. ii. (a) Express the Parametric Curve by an Equation in iii. x and y: iv. i. v. ii. vi. iii. (d) Calculate the Area enclosed by the Polar Curve: iv. v. i. ii. (b) Find a Parametrization x = (t), y = y(t), t ∈ [0, 1] for iii. i. iv. ii. v. iii. (e) Find the Slope of the Polar Curve: iv. i. (c) ii. (d) Find the slope of the given curve at the given iii. point and give an equation of the tangent line: (f) Find the Length of the Polar Curve: i. i. ii. ii. 3

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differential and integral calculus, differential equations, differential calculus, integral calculus, fourier transform, fourier series, first order, taylor's theorem, homework problems, how to, bessel functions, elementary functions, boundary conditions, functions of several variables, definite integral

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posted: | 7/29/2010 |

language: | English |

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