In Problems 21–26, use the description of the region R to evaluate the indicated
24.
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- 3. Use the graph and the actual areas of the indicated regions to evaluate the integrals in the following problems. a. b. C. d f(x) dx = f(x) [*1x) dx = 5 'b d f(x) dx = f(x) A a B b с \y = f(x) Area A = 1 Area B = 2 Area C = 2 →x Area D = 0.6arrow_forward9. If f(x) is a continuous functions, which of the following integrals should have the same value? I. . | f(x)dx .b-a II. Sf(x + a)dx .b+c III. f(x + c)dx ate (A) I and II only (в) I and II only (c) II and III only (D) I, II, and III (E) None of the abovearrow_forward2. Let D be the region bounded by (0 <1< 1), (b) y = -r + 2x (1<< 2), (c) y = x - 2 (1<< 2), (d) y = -r2 (0 << 1). (a) y = I Explain in detail how to compute / - y) dz dy in three different ways. For two of the ways try the D change of variables z = u +v and y = v-u².arrow_forward
- 40. Evaluate O (x2 – 2xy) dx + (2y +3)dy around the boundary of the region defined by y? = 8x and x = 2 (a ) directly, (b) by using Green's the orem. Ans. 128/5arrow_forward14. Evaluate the integral: (x² + 3x)° dx A. (x* + 30)' + C 3 x* + 3x' + C C. ( + 3x) + C В. +C D.arrow_forward3. SHOW YOUR COMPLETE/DETAILED/CORRECT SOLUTION.arrow_forward
- 4. Evaluate the double integral LL -Y by using change of variables +2y+3 x + y (x - 2y)² x-2y dx dy, u= x - 2y, v=x+y.arrow_forward16.10 f(x, y) = 4x + 7y and D = {(x, y)|0 ≤ x ≤ 1, x³ ≤ y ≤ x³ + 1} Evaluate the double integral f(x, y) dA over the region D.arrow_forward2. Evaluate each of the following line integrals in two ways*. F (x, y) = (2x – cos y) i+ (x sin y) and C1 is the straight-line path from (-4,0) to (0,5). (a) | F dr, where C1 F-(x, y) = (2x – y) i + (2x + y) j and C2 is a circle of radius 4 centered at the origin, traversed once counterclockwise starting at (4,0). (b) / F2· dĩ, where F2 · dr, where C2 Acceptable ways to evaluate the integral: * • Directly: Parametrize the path and write the integral in terms of your parametrization. • Using the Fundamental Theorem of Line Integrals: Write the theorem and show what you substitute for each part. • Using Green's Theorem: Write the theorem and show what you substitute for each part.arrow_forward
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