Weekly Tutorial 6 for Math. 69.107A,
Elementary Calculus I

1. Evaluate the definite integral of the function f(x) = e^{3 x3} x² over the x-interval ranging from 0 to 1.

a) (e³ - 2)/9
b) 0
c) (e³ - 1)/9
d) e³/9
e) None of these

2.Find the area between the curves f(x) = sin(x) and g(x) = cos(x) for x in [Pi/4, 5 Pi/4].

a) 2 sqrt(2)
b) sqrt(2)
c) 0
d) 3 sqrt(2)
e) None of these

3.Evaluate the definite integral of the function f(x) = (Arcsin(t))²/sqrt(1-t²) between the limits 0 and 1.

a) Pi³/18
b) Pi³/24
c) Pi³/8
d) Pi³
e) None of these

4. Evaluate the indefinite integral of the function f(t) = sin(t) cos(t)/(1-sin²(t)).

a) ln(sec(t)) + C
b) ln(cos(t)) + C
c) ln(csc(t)) + C
d) ln(sin(t)) + C
e) None of these

5.Evaluate the indefinite integral of the function cot³(u) csc⁴(u).

a) - (cot⁴(u))/4 - (cot⁶(u))/6 + C
b) - (cot³(u))/3 - (cot⁵(u))/5 + C
c) - (cot⁴(u))/4 + C
d) - (cot⁵(u))/5 + C
e) None of these

6. Evaluate the area under the curve y = cos³(x) where x is in [0, Pi/2].

a) 1
b) 1/3
c) 2/3
d) 4/3
e) None of these

Check your answers by clicking the adjoining box: 1. c ...Let u = 3 times x cubed.... 2. a ...Evaluate the definite integral 3. b ...Let u = Arcsin(t) ... 4. a ...Use an identity in the denominator and then let u = cos(t) 5. a ...Write the second term as csc(u) squared squared and simplify one of them using an identity. Then let u = cot(u). 6. c ...Evaluate the definite integral. Use an identity relating sin & cos to the square of cos. ...