Question

Group Activity Making Connections Suppose that $$\int f(x) d x=F(x)+C$$ (a) Explain how you can use the derivative of \(F(x)+C\) to confirm the integration is correct. (b) Explain how you can use a slope field of \(f\) and the graph of \(y=F(x)\) to support your evaluation of the integral. (c) Explain how you can use the graphs of \(y_{1}=F(x)\) and \(y_{2}=\int_{0}^{x} f(t) d t\) to support your evaluation of the integral. (d) Explain how you can use a table of values for \(y_{1}-y_{2}\) \(y_{1}\) and \(y_{2}\) defined as in part (c), to support your evaluation of the integral. (e) Explain how you can use graphs of \(f\) and \(\mathrm{NDER}\) of \(F(x)\) to support your evaluation of the integral. (f) Illustrate parts (a)- (e) for \(f(x)=\frac{x}{\sqrt{x^{2}+1}}\) .

Short answer

In brief, after integrating \(f(x)\) to get \(F(x) + C\), to verify the integration: (a) differentiate \(F(x) + C\) and it should give \(f(x)\), (b) the slope field of \(f(x)\) should follow the curve of \(F(x)\), (c) the graph of \(\int_{0}^{x} f(t) dt\) should coincide with \(F(x)\), apart from a constant, (d) the difference \(F(x) - \int_{0}^{x} f(t) dt\) should be a constant, as seen in a table of values, (e) \(NDR(F(x))\) and \(f(x)\) must coincide. If the given function \(f(x) = \frac{x}{\sqrt{x^{2}+1}}\), each of the steps from (b) through (e) could be illustrated graphically to support the evaluation of the integral.

Step-by-step solution

Explanation for part (a)

The process of differentiation is the reverse of integration; therefore if you take the derivative of \(F(x)+C\) and it equates to \(f(x)\), then the integration is correct.

Explanation for part (b)

The slope field of \(f(x)\) represents the derivative of \(F(x)\) at each point on the graph. A solution curve of \(F(x)\) should follow the trends of the slope field. If it does, then it's likely the integral has been evaluated correctly.

Explanation for part (c)

The graphs of \(y_{1}=F(x)\) and \(y_{2}=\int_{0}^{x} f(t) d t\) should be identical apart from the constant of integration. This is due to the second graph measuring the net area under the function up to point x, which is the definition of an antiderivative.

Explanation for part (d)

Values of \(y_{1}-y_{2}\) on the table should remain constant since \(y_{1}=F(x)\) and \(y_{2}=\int_{0}^{x} f(t) d t\) differ by a constant only, this constant representing area under the curve from 0 up to x. If this remains true in the table of values, then the integral is evaluated correctly.

Explanation for part (e)

The graph of the function \(NDR\) of \(F(x)\) is \(f(x)\), due to the fact that \(NDR\) denotes the derivative. \(f(x)\) and \(NDR(F(x))\) must therefore coincide.

Applying to given function \(f(x)\)

For example, if \(f(x)=\frac{x}{\sqrt{x^{2}+1}}\), then integral is \(F(x)=\sqrt{x^2+1}+C\). Part (a) can be illustrated by deriving \(F(x)\), which indeed yields \(f(x)\). The rest of the subparts (b) through (e) can be illustrated by plotting the function \(f(x)\), its integral \(F(x)\), and slope fields or integral from 0 to x as appropriate.