35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of

Find the area of the region R.

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Identify the region R in the first quadrant bounded by the piecewise function \( f(x) \) and the x-axis. The function is given by: \[ f(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 2 - \sqrt{x} & \text{if } 1 < x \leq 4 \end{cases} \] This means the region is bounded above by \( f(x) \) and below by the x-axis from \( x=0 \) to \( x=4 \).

To find the area of region R, set up the integral of \( f(x) \) over the interval \( [0,4] \). Since \( f(x) \) is piecewise, split the integral into two parts: \[ \text{Area} = \int_0^1 1 \, dx + \int_1^4 (2 - \sqrt{x}) \, dx \]

Evaluate the first integral: \[ \int_0^1 1 \, dx = [x]_0^1 = 1 - 0 \] This represents the area of the rectangle under \( f(x) = 1 \) from 0 to 1.

For the second integral, rewrite the integrand and integrate term-by-term: \[ \int_1^4 (2 - \sqrt{x}) \, dx = \int_1^4 2 \, dx - \int_1^4 x^{1/2} \, dx \] Calculate each integral separately using the power rule for integration.

After finding the antiderivatives, apply the Fundamental Theorem of Calculus by evaluating the definite integrals at the limits \( x=1 \) and \( x=4 \). Then, sum the results of both integrals to get the total area of region R.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Piecewise Functions

A piecewise function is defined by different expressions over distinct intervals of the domain. Understanding how to interpret and work with these functions is essential, especially when calculating areas or integrals, as each piece may require separate treatment.

Definite Integrals for Area Calculation

The definite integral of a function over an interval gives the net area between the function's graph and the x-axis. For regions bounded by piecewise functions, the total area is found by summing integrals over each sub-interval where the function is defined differently.

Square Root Functions and Their Integration

Functions involving square roots, such as √x, require careful integration techniques. Knowing how to rewrite and integrate expressions like 2 - √x is crucial for accurately finding areas under such curves.

35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of Find the area of the region R.