Ch. 5 - Integrals

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 5 - Integrals

Problema 5.PE.13

Capitolo 5, Problema 5.PE.13

In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.
ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8

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First, understand that the total area between the graph of the function and the x-axis means we need to find the integral of the absolute value of the function over the interval \(-1 \leq x \leq 8\). This is because areas below the x-axis contribute positively to the total area when taken as absolute values.

Next, identify where the function \(f(x) = 5 - 5x^{2/3}\) intersects the x-axis by solving \(f(x) = 0\). Set \(5 - 5x^{2/3} = 0\) and solve for \(x\) to find the points where the graph crosses the x-axis.

Once the zeros are found, split the integral into subintervals where the function is either positive or negative. For each subinterval, set up the definite integral of \(|f(x)|\) which will be either \(f(x)\) or \(-f(x)\) depending on the sign of \(f(x)\) in that interval.

Write the total area as the sum of the integrals over these subintervals: \(\text{Area} = \int_{a}^{b} |f(x)| \, dx = \sum \int_{x_i}^{x_{i+1}} |f(x)| \, dx\), where \(x_i\) are the zeros and the endpoints \(-1\) and \(8\).

Finally, compute each integral separately by integrating \(f(x)\) or \(-f(x)\) as appropriate, using the power rule for integration on \(x^{2/3}\), and then sum the absolute values of these integrals to get the total area.

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Definite Integral and Area Under a Curve

The definite integral of a function over an interval represents the net area between the graph and the x-axis. Positive areas above the x-axis add to the integral, while areas below subtract. To find total area, consider absolute values of areas where the function is negative.

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05:43

Definition of the Definite Integral

Identifying Points Where the Function Crosses the x-axis

To find total area between the curve and x-axis, first determine where the function equals zero within the interval. These points divide the region into subintervals where the function is either positive or negative, allowing separate integration and correct area calculation.

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Determining Where a Function is Increasing & Decreasing Example 1

Handling Functions with Fractional Exponents

Functions with fractional exponents, like x^(2/3), require careful evaluation, especially near zero or negative inputs. Understanding how to differentiate and integrate such functions is essential, as well as recognizing their domain and behavior to correctly set up integrals.

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