CSUN MATH 150A: Practice Problem Set on the Cumulative
Effect of a Function (Integration)
Section I: Conceptual Understanding and Riemann Sums
This section focuses on approximating the cumulative effect using geometry. Show
your work, including the calculation for Δ x and the summation formula.
Problem 1: Approximating Area with Rn and Ln
Consider the function f ( x)=x2 +1 on the interval [0 , 6].
1. Calculate Δ x for n=3 subintervals.
2. Estimate the area under the curve using the Right Endpoint Riemann Sum
( R3).
3. Estimate the area under the curve using the Left Endpoint Riemann Sum (
L3).
4. Is R3 an overestimate or an underestimate? Justify your answer based on
the functions behavior (increasing/decreasing).
Problem 2: Midpoint Riemann Sum and Summation Notation
Consider the function g( x )=4 x − x 2 on the interval [1, 5].
1. Calculate Δ x for n=4 subintervals.
2. Determine the midpoint x ∗i for each of the four subintervals.
3. Estimate the area under the curve using the Midpoint Riemann Sum ( M 4 ).
4. Write the expression for the general Right Endpoint Riemann Sum ( Rn ) for
g( x ) over [1, 5] using sigma notation.
Problem 3: Interpreting the Definite Integral
A particles velocity, in meters per second, is modeled by v (t)=3 t 2 − 6 t .
1. Write a definite integral expression that represents the net change in
position (displacement) of the particle from t=0 to t=4 seconds.
2. Write a definite integral expression that represents the total distance
traveled by the particle from t=0 to t=4 seconds.
, 3
3. Explain the difference between the physical meaning of ∫ v (t) dt and
1
3
∫ ¿ v (t)∨dt .
1
Section II: Fundamental Theorem of Calculus Part II (Computational)
b
Use the FTC Part II (∫ f ( x )dx=F (b) − F (a)) to evaluate the following definite
a
integrals.
Problem 4: Polynomial and Power Rule
Evaluate:
3
∫ (¿ 6 x 2 − 4 x +5)dx ¿
1
Problem 5: Rational and Logarithmic Functions
Evaluate:
2
e
∫
e
( 1 1
−
x x2)dx
Problem 6: Trigonometric Functions
Evaluate:
π /4
∫ (¿ 3 sin( x )+sec2 ( x ))dx ¿
0
Problem 7: Piecewise Functions
Evaluate the definite integral for the piecewise function f (x):
{
3 2
4 x if x< 1
f (x)= ∫ f (x) dx
5 − x if x ≥ 1 0
Problem 8: Exponential Functions
Evaluate:
Effect of a Function (Integration)
Section I: Conceptual Understanding and Riemann Sums
This section focuses on approximating the cumulative effect using geometry. Show
your work, including the calculation for Δ x and the summation formula.
Problem 1: Approximating Area with Rn and Ln
Consider the function f ( x)=x2 +1 on the interval [0 , 6].
1. Calculate Δ x for n=3 subintervals.
2. Estimate the area under the curve using the Right Endpoint Riemann Sum
( R3).
3. Estimate the area under the curve using the Left Endpoint Riemann Sum (
L3).
4. Is R3 an overestimate or an underestimate? Justify your answer based on
the functions behavior (increasing/decreasing).
Problem 2: Midpoint Riemann Sum and Summation Notation
Consider the function g( x )=4 x − x 2 on the interval [1, 5].
1. Calculate Δ x for n=4 subintervals.
2. Determine the midpoint x ∗i for each of the four subintervals.
3. Estimate the area under the curve using the Midpoint Riemann Sum ( M 4 ).
4. Write the expression for the general Right Endpoint Riemann Sum ( Rn ) for
g( x ) over [1, 5] using sigma notation.
Problem 3: Interpreting the Definite Integral
A particles velocity, in meters per second, is modeled by v (t)=3 t 2 − 6 t .
1. Write a definite integral expression that represents the net change in
position (displacement) of the particle from t=0 to t=4 seconds.
2. Write a definite integral expression that represents the total distance
traveled by the particle from t=0 to t=4 seconds.
, 3
3. Explain the difference between the physical meaning of ∫ v (t) dt and
1
3
∫ ¿ v (t)∨dt .
1
Section II: Fundamental Theorem of Calculus Part II (Computational)
b
Use the FTC Part II (∫ f ( x )dx=F (b) − F (a)) to evaluate the following definite
a
integrals.
Problem 4: Polynomial and Power Rule
Evaluate:
3
∫ (¿ 6 x 2 − 4 x +5)dx ¿
1
Problem 5: Rational and Logarithmic Functions
Evaluate:
2
e
∫
e
( 1 1
−
x x2)dx
Problem 6: Trigonometric Functions
Evaluate:
π /4
∫ (¿ 3 sin( x )+sec2 ( x ))dx ¿
0
Problem 7: Piecewise Functions
Evaluate the definite integral for the piecewise function f (x):
{
3 2
4 x if x< 1
f (x)= ∫ f (x) dx
5 − x if x ≥ 1 0
Problem 8: Exponential Functions
Evaluate:
