Find the area enclosed by the curves and find the area of

Question

I hope and expect to be asked questions related to the homework. However, I will not provide a solution to a problem before it is due. Uncertainty, confusion, and frustration are signs you're working to make sense of the material-they're normal and valuable parts of the learning process. You should work hard to solve each problem on your own, as your mastery of the material ultimately depends on you and your desire to participate.

Provide precise and thoughtful solutions to the following problems. How you reached your answer is more important than the answer itself. Also, remember that many problems are best understood with a graph or a picture.

1. Evaluate the following integrals.

1. a ₁∫²⁷ ((t + 1)/√t).dt

1.b ₀∫^Π/2cos (1/3x) dx

1.c _-1∫^e 1/x dx

1.d ₂∫⁶ x + 1/x dx

2. Let A(x) = _-2∫^x√(t²+1)dt

Calculate:

2.a A(-2)
2.b A'( -2)

3. Let A(x) = ₀∫^xf(t)dt, where the graph of f is given below.

Determine:

3.a the intervals on which A is increasing and decreasing

3.b the values of x where A has a local minimum or local maximum

3.c the inflection points of A

3.d the intervals where A is concave up or concave down

4. Determine f (x), given that ₀∫^x f(t) dt = xe^2x cos(5x).

5. Evaluate the following integrals.

5.a ∫sin²(x) cos x dx

5.b ∫ (lnx)²/x dx

5.c ∫ sec(x) tan(x) (sec(x) - 1) dx

6. Evaluate the following integrals, which are related to transcendental functions.

6.a ₀∫³ dx/(x² + 3)

6.b ∫ x/(x⁴ + 1)dx

7. Find a > 0 so that the area between f(x) = a/a²+x² and the x-axis for 0 ≤ x ≤ 1 is equal to Π/3.

8. Evaluate ∫x²√(x -5) dx.

9. Evaluate ∫dx/((3x +1)ln(6x +2)). Hint: It's possible to use the method of substitution once or twice.

10. Find the value of the constant c so that the region lying between y = e^x and the x-axis for 0 ≤ x ≤ c has the same area as the region between y = 3^x and the x-axis for 0 ≤ x ≤ 3.

11. Find the area enclosed by the curves f(x) = c - x² and g(x) = x² - c as a function of c (i.e., your answer will depend on c). Then find the value of c for which this area is equal to 1.

12. Find the area enclosed by the curves f(x) = 4x³ - 3x and g(x) = 5x - 4x³.

13. Find the area of the region bounded by f(x) = sin x and g(x) = 2/Πx. Hint: The graphs intersect at three points.