Overview
In this section we see how antiderivatives allow us to compute a definite integral using the The Fundamental Theorem of Calculus. In this result, we see formally how the net-signed area under a curve is connected to the antiderivative of the function that generates the curve. This result extends our earlier work where we saw that slopes on the graph of \(f\) generate heights on the graph of \(f'\); now we can also see that net-signed areas between \(f'\) and the \(x\)-axis are connected to differences in heights on \(f\).
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Understand that the change in position of an object gives the net-signed area bounded by a velocity curve on an interval \([a,b]\): \(\int_a^b v(t)\;dt = s(b)-s(a)\).
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Define what an antiderivative is.
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Understand how the previous result can be generalized to give the Fundamental Theorem of Calculus: \(\int_a^b f(x)\;dx = F(b) - F(a)\), where \(F\) is an antiderivative of \(f\).
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Apply the Fundamental Theorem of Calculus to compute definite integrals where the integrand is a sum or diffence of terms with power or exponential functions.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Compute the family of antiderivatives of a function and undertand why two antiderivatives of the same function only differ by a constant (be able to show on a graph why it is so, or use theorems from differential calculus to support your claim).
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State the Total Change Theorem and understand how it is simply a restatement of the Fundamental Theorem of Calculus.
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Apply the Total Change Theorem to solve different applied problems.
To prepare for class
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Read subsection 4.4.1 in Active Calculus: The Fundamental Theorem of Calculus. This theorem generalizes the formula \(\int_a^b v(t)\;dt = s(b)-s(a)\).
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Watch the following video (by GVSUmath) on the Fundamental Theorem of Calculus:
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Read subsection 4.4.2 in Active Calculus: Basic antiderivatives.
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Watch the following video (by GVSUmath) which shows how we apply the Fundamental Theorem of Calculus when the function to be anti-differentiated contains power functions. Note that the author forgets to put equality signs to the left of her expressions, each time she changes line:
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Watch the following video (by GVSUmath) which shows how we apply the Fundamental Theorem of Calculus when the function to be anti-differentiated contains exponential functions:
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
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Read subsection 4.4.3 in Active Calculus: The total change theorem
