Marianopolis · 201-AS3-MS

Marianopolis 201-AS3-MS: Integral Calculus Tutoring

A tutor can help you choose an integration method, work through unclear steps and interpret your result.

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The integral calculus course

201-AS3-MS is the Integral Calculus course in Marianopolis Arts & Sciences. It develops integration methods and applications, with improper integrals and an introduction to series.

Course topics

Definite and indefinite integrals; the Fundamental Theorem of Calculus
Integration methods
Area, volume and differential-equation applications
Indeterminate forms and improper integrals
Introductory series and power series

Prepare for integration

Practise reversing familiar derivative rules, simplifying algebra and finding where a function changes sign. These skills help you check both an antiderivative and the region an integral describes.

Two points to practise

Choosing a method and checking the antiderivative by differentiation
Distinguishing signed accumulation from total geometric area

Signed integral or total area: choose what to add

For $f(x)=x-1$ on $[0,3]$ , calculate $\int_0^3 f(x)\,dx$ and the total area between the graph and the x-axis. Before integrating, sketch the line, mark its zero and predict whether the two answers should agree. Coordinates are dimensionless; report geometric areas in square coordinate units.

Show the worked solution

Evaluate the definite integral
An antiderivative is $F(x)=x^2/2-x$ . Thus $\begin{gathered}\int_0^3(x-1)\,dx\\=F(3)-F(0)\\=3/2\end{gathered}$ .
Split at the sign change
The graph crosses the x-axis at $x=1$ . It is below the axis on $[0,1)$ and above it on $(1,3]$ . The total area is $\begin{gathered}-\int_0^1(x-1)\,dx\\{}+\int_1^3(x-1)\,dx\\=1/2+2=5/2\end{gathered}$ .
The definite integral counts the first region negatively; total area counts both regions positively. A correct antiderivative is only part of the solution: the quantity requested determines how to use it.

Retry with a different zero and interval

Let $h(x)=x-2$ on $[0,5]$ . Find its signed integral and the total area between its graph and the x-axis. Write the split point before calculating. Would taking the absolute value of the signed integral give the total area? Coordinates remain dimensionless.

Check the retry

The zero is $x=2$ . Using $H(x)=x^2/2-2x$ , the contributions are $H(2)-H(0)=-2$ and $H(5)-H(2)=9/2$ . The signed integral is $-2+9/2=5/2$ . Total area is $2+9/2=13/2$ square coordinate units.

Taking $|5/2|$ still gives $5/2$ : taking the absolute value after adding does not undo the cancellation between the two signed contributions. For total area, take the magnitude of each contribution before adding.

Work on the step that changed the answer

If the split point was wrong, solve $h(x)=0$ and mark the signs on a sketch. If the signed contributions were wrong, check the antiderivative and subtract the lower-bound value. If only total area was wrong, explain which region must count positively. A tutor can review those three decisions with your sketch, then give you a different crossing point to handle independently.

For another area question, try question 7 in the integration practice. That set has a Concordia MATH 205 context; use this question as additional topic practice, not as an official Marianopolis assessment.

Work with a tutor

A tutor can help you choose an integration method, explain the algebra and work through a first example. If you have already started a solution, you can review it together. For application problems, you can sketch the situation, build the bounds together and discuss what the result measures. Bring a practice question, even if you do not know how to begin.

Plan your review

Choose a problem from the chapter you are studying. Before calculating, name the quantity requested and explain your method. Afterward, check any antiderivative by differentiating it, then review your bounds and signs. Try a similar problem without the worked solution.

Independent practice

Choose your next practice step

35 min

How to Choose an Integration Method

Compare substitution, integration by parts and partial fractions.

40 min

Volumes of Revolution: Build the Cross-Section First

Work from a sketch to the cross-sectional area and integral.

35 min

Improper Integrals: Define the Limit, Decide Convergence

Practise defining the limit for an infinite interval or an unbounded integrand.

Course description

Marianopolis course description

Frequently asked questions

Which language and platform do lessons use?

Tutoring sessions are currently in English, online through Zoom or Google Meet.

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