Ch. 8 - Techniques of Integration

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

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problema 8.2.58a

Capitolo 8, Problema 8.2.58a

Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
a. π/2 ≤ x ≤ 3π/2.

Guida verificata passo dopo passo

Identify the function and the interval: The function given is $y = x \cos x$, and the interval is $\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}$.

Analyze the graph to determine where the function is above or below the x-axis in the given interval. From the graph, $y = x \cos x$ is negative between $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ because $\cos x$ is negative in this interval and $x$ is positive.

Since the function is below the x-axis in this interval, the area enclosed by the curve and the x-axis is the integral of the absolute value of the function. This means we need to integrate $-x \cos x$ over $\left[ \frac{\pi}{2}, \frac{3\pi}{2} \right]$.

Set up the integral for the area: $\text{Area} = \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} -x \cos x \, dx$.

To solve the integral, use integration by parts where you let $u = x$ and $dv = \cos x \, dx$. Then compute $du = dx$ and $v = \sin x$. Apply the integration by parts formula: $\int u \, dv = uv - \int v \, du$.

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Definite Integral for Area Calculation

The definite integral of a function over an interval gives the net area between the curve and the x-axis. When the function crosses the x-axis, the integral sums positive and negative areas, so the absolute value or splitting the integral is needed to find the total enclosed area.

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Definition of the Definite Integral

Behavior of the Function y = x cos(x)

The function y = x cos(x) oscillates and crosses the x-axis at points where cos(x) = 0, such as x = π/2 and 3π/2. Understanding these zeros helps determine the limits for integration and whether the function is above or below the x-axis in the given interval.

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Integrals of Natural Exponential Functions (e^x)

Splitting the Integral at Zeros of the Function

To find the total area enclosed by the curve and the x-axis when the function changes sign, split the integral at the zeros of the function. Calculate the integral over each sub-interval and take the absolute value before summing to get the total area.

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Integrals of General Exponential Functions

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