Chapter 4: Integration

Section 4.3: Fundamental Theorem of Calculus and the Indefinite Integral

Example 4.3.2

$$ Use the FTC to evaluate the definite integral ${\int }_{-1}^3\left(5 x^3-7 x^2\+4\right) \mathit{\ⅆ}x$. $$ Solution $$ •  Control-drag the definite integral. •  Context Panel: Evaluate and Display Inline ${\int }_{-1}^3\left(5 x^3-7 x^2\+4\right) \mathit{\ⅆ}x$ = $\frac{152}{3}$$$$$ $$ A solution from first principles requires more arithmetic than calculus. Once the antiderivative is found by application of the Power rule, several lines of tedious arithmetic ensue. If an error is going to be made in this calculation, it will most likely be arithmetic.   ${\int }_{-1}^3\left(5 x^3-7 x^2\+4\right) \mathit{\ⅆ}x$ $\={\left[5 \frac{x^4}{4}-7 \frac{x^3}{3}\+4 x\right]}_{-1}^3$ $$ $\=\left(\frac{5}{4} 3^4-\frac{7}{3} 3^3\+4\cdot 3\right)-\left(\frac{5}{4}{\left(-1\right)}^4-\frac{7}{3}{\left(-1\right)}^3\+4\left(-1\right)\right)$ $$ $\=3\left(\frac{5}{4}\cdot 27-21\+4\right)-\left(\frac{5}{4}\+\frac{7}{3}-4\right)$ $$ $\=3\cdot \frac{67}{4}-\left(-\frac{5}{12}\right)$ $$ $\=\frac{9\cdot 67\+5}{12}$ $$ $\=\frac{152}{3}$ $$

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