MA 114 Worksheet # 17: Integration by trig substitution

Transcription

1 MA Worksheet # 7: Integration by trig substitution. Conceptual Understanding: Given identity sin θ + cos θ =, prove that: sec θ = tan θ +. Given x = a sin(θ) with a > and π θ π, show that a x = a cos θ. (c) Given x = a tan(θ) with a > and π < θ < π, show that a + x = a sec θ. Given x = a sec(θ) with a > and θ < π or π θ < 3π, show that x a = a tan θ.. Compute following integrals: (c) (e) u 3 6 u du x 5 x dx x 3 dx 6 + x x + dx x x + dx 3. Let a, b >. Prove that area enclosed by ellipse x a + y = is πab. b. Let r >. Consider identity s r x dx = r arcsin (s/r) + s r s where s r. Plot curves y = r x, x = s, and y = x r s s. Using part, verify identity geometrically. (c) Verify identity using trigonometric substitution.

2 MA Worksheet # 8: Method of Partial Fractions and Numerical Integration. Write out general form for partial fraction decomposition but do not determine numerical value of coefficients. x + 3x + x + x + x + x (c) (x + )(x + )(x + ) x + 5 (x + ) 3 (x + ). Compute following integrals. x 9 (x + 5)(x ) dx x + 3x + dx x 3 x + (c) x 3 x dx x 3 + x + dx (e) x(x + ) dx 3. Compute x 3 x dx by first making substitution u = 6 x.. Conceptual Understanding: Write down Midpoint rule and illustrate how it works with a sketch. Write down Trapezoidal Rule and error bound associated with it. 5. Use Midpoint rule to approximate value of if approximation is an overestimate or an underestimate of integral. e x dx with n =. Draw a sketch to determine 6. The velocity in meters per second for a particle traveling along axis is given in table below. Use Midpoint rule to approximate total distance particle traveled from t = to t = 6. t v(t)

3 MA Worksheet # 9: Numerical Integration. Simpson s Rule turns out to exactly integrate polynomials of degree three or less. Show that Simpson s rule gives exact value of h p(x) dx where h > and p(x) = ax 3 + bx + cx + d. [Hint: First compute exact value of integral by direct integration. Then apply Simpson s rule with n = and observe that approximation and exact value are same.]. Use Midpoint Rule and n Simpson s Rule to approximate integral Compare your results to actual value to determine error in each approximation. 3. Use Trapezoid Rule, Midpoint Rule and Simpson s Rule to approximate integral with n = 8. π x sin x dx with n = 8. + x dx

4 MA Worksheet # : Arc Length and Surface Area. Conceptual Understanding: Write down formula for arc length of a function f(x) over interval [a, b] including required conditions on f(x). Write down formula for (surface) area of surface obtained by rotating graph of f(x) about x-axis for a x b. How would this formula change if graph were instead rotated about y = c?. Find an integral expression for arc length of following curves. Do not evaluate integrals. f(x) = sin(x) from x = to x =. f(x) = x from x = to x = 6. (c) x + y = 3. Find arc length of following curves. f(x) = x 3/ from x = to x =. f(x) = ln(cos(x)) from x = to x = π/3.. Set up a function s(t) that gives arc length of curve f(x) = x + from x = to x = t. Find s(). 5. Calculate arc length of f(x) = x over [, ]. [Hint: You will need to use a trigonometric substitution.] 6. Calculate arc length of graph of f(x) = mx + r over [a, b] in two ways: using Pythagorean Theorem and using arc length integral. [Hint: Make arc of f(x) = mx + r from [a, b] hypotenuse of a right triangle with legs (b a) and m(b a).] 7. Use Simpson s Rule with n = 6 to approximate arc length of f(x) = sin(x) over [, π]. For Problems 8, compute surface area for a revolution about x-axis over given interval. 8. y = x, [, ] 9. y = x 3, [, ]. y = ( x /3) 3/, [, 8]

5 ... Conceptual Understanding: MA MA Worksheet # : 3: Center of Mass Write down formulas for coordinates of centroid of a plate with constant density bounded between x = a, x = b, f(x), Write down formulas for coordinates of centroid of a and plate g(x) with as in constant figure density to bounded right. between x = a, x = b, f(x), and andg(x) as asinin figure to to right. a g(x) f(x) bb Write down formulas for coordinates of centroid of a plate Write Write withdown constant formulas density for bounded for coordinates between y of = c, y centroid = d, f(y), of a and plate g(y) platewith as inconstant figure density to bounded right. between y = c, y = d, f(y), and andg(y) g(y) as asinin figure figuretoto right. d c g(y) g(y) f(y) f(y). Find center of mass for system particles, at coordinates. Find. Find center center of of mass mass for for system of ofparticles of ofmasses,,,, 5, and located at coordinates (, ), ( 3, ), (, ), and (, ). (, (, ), ), ( 3, ( 3, ), ), (, (, ), ), and and (, (, ). ). 3. Point masses of equal size are placed at vertices of triangle with coordinates (3, ), (b, ), and 3. Point 3. Point (, masses masses 6), where of of equal equal b > 3. size size Find are are placed placed center at at of mass. vertices of of triangle with coordinates (3, ), (b, ), ), and (, (, 6), 6), where where b > b 3. > 3. Find Find center center of of mass. mass.. Find centroid of region under graph of for. For practice, do this using. Find. Find both centroid centroid approach of of from region region under under and approach graph graph of of from y y =. x for for x.. For practice, do do this this using using both both approach approach from from and and approach approach from from.. 5. Find centroid of region under graph of f(x) for. 5. Find centroid of region under graph of f(x) for. 5. Find centroid of region under graph of f(x) = x for x. 6. Find centroid of region between f(x) = x and g(x) for. 6. Find centroid of region between f(x) = x and g(x) for. 6. Find centroid of region between f(x) = x 7. Let m > n. Find centroid of region and g(x) between x m = and n x for x. for x. Find values for m 7. Let and m > that n force. Find centroid centroid to lie ofoutside region of between region. x m and n for. Find values for 7. Let m > n. Find centroid of region between x and n that force centroid to lie outside of region. m and x n for x. Find values for m and n that force centroid to lie outside of region.

6 MA Worksheet # : Differential Equations and y = k(y b). Conceptual Understanding: What does it mean to say that a differential equation is first-order (or second-order or thirdorder... ) What does it mean to say that a differential equation is linear or nonlinear?. Use Separation of Variables to find general solutions to following differential equations. y + xy = x y = xy (c) ( + x )y = x 3 y + y y + sec x = 3. Solve y = y + subject to condition that y() = 5.. Solve y + 6y = subject to condition that y() =. 5. Recall that Newton s law of Cooling stipulates that temperature y(t) of a cooling object with respect to time satifies differential equation y = k(y T ), where k is a constant depending on object and T is temperature of ambient environment. Frank s car engine runs at F. On a 7 F day, he turns off ignition and notes that five minutes later, engine has cooled to 6 F. Find cooling constant k. When will engine cool to F? 6. A cup of coffee with cooling constant k =.9min is placed in a room of temperature C. How quickly is coffee cooling when tempurature is 8 C? Use linear approximation to estimate change in temperature over next 6 s when temperature is 8 C. (c) If coffee is initially served at 9 C, how long will it take to reach an optimal drinking temperature of 65 C? 7. (Extra) A tank has shape of parabola y = x revolved about x-axis. water leaks from a hole of area B =.5 m at bottom of tank. Let y(t) be water level at time t. How long does it take for tank to empty if initial water level is y() = m?

7 MA Worksheet # 3: Graphical Methods. Match differential equation with its slope field. Give reasons for your answer. y = y y = x( y) y = x + y y = sin(x) sin(y) Slope field I Slope field II (c) Slope Field III Slope field IV Figure : Slope fields for Problem. Use slope field labeled IV to sketch graphs of solutions that satisfy given initial conditions y() =, y() =, y() =. 3. Sketch slope field of differential equation. Then use it to sketch a solution curve that passes through given point y = y x, (, ) y = xy x,(, ). Show that isoclines of y = t are vertical lines. Sketch slope field for t, y and plot integral curves passing through (, ) and (, ).

8 MA Worksheet # : Review for Exam 3. Power, Maclaurin, and Taylor Series x Find Maclaurin series for +x. Find Taylor series for cos x about a = π/.. Compute dx x x (c) 6x +8 x +9 dx 3 x (x +)(x + x) dx + x +3 dx e x 3. Compute e x e x dx. Hint: First do a substitution, and n use partial fractions. dx. Evaluate x first with a trig substitution and n with partial fractions. Verify that answer is same in both cases. 5. Recall Trapezoid, Midpoint and Simpson s Rule. Compute M and T for Compute T and S for x dx x dx. 6. An airplane s velocity is recorded at 5 minute intervals during a hour flight with following results, in miles per hour: Estimate total distance traveled by plane during hour using Simpson s Rule. {55, 575, 6, 58, 6, 6, 65, 595, 59, 6, 6, 6, 63} 7. Find arc length of f(x) = ln(sec(x)) from x =tox = π/. 8. Find surface area of solid of revolution obtained by revolving 9 x about x-axis for x. 9. Consider point masses m, m,andm 3 centered at (, ), (3, ), and (, ) respectively. If m =6, find m so that center of mass lies on y-axis.. Use separation of variables to solve y +xy =.. Use separation of variables to solve y =(x +)(y +).. Find solutions to y = y + 8 subject to y() = 3 and y() =, respectively, and sketch ir graphs.

9 3. Match each of slope fields below with exactly one of differential equations. (The scales on x- and y-axes are same.) Also, provide enough explanation to show why no or matches are possible. (i) y = xy + (ii) y = xe y (iii) y = y + (iv)y =siny (c) Figure : Slope fields for Problem 3