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Magazine Chapter 5: Problem 73
Find the area of the region in the first quadrant bounded by the line \(y=x,\) the line \(x=2,\) the curve \(y=1 / x^{2},\) and the \(x\) -axis.
Short Answer
Expert verified
The area is 3 square units.
Step by step solution
01
Identify the region
We need to sketch the bounds defined by the given equations in the first quadrant: the line \(y = x\), the vertical line \(x = 2\), the curve \(y = \frac{1}{x^2}\), and the \(x\)-axis. These bounds form a region in the first quadrant of the \(xy\)-plane.
02
Set up the integral for the area
The area can be found by integrating the vertical distance from the curve \(y = \frac{1}{x^2}\) to the line \(y = x\) over the interval \([1, 2]\), which is the region of intersection. Since \(y = x\) is above \(y = \frac{1}{x^2}\) in this range, the integral is: \[ A = \int_{1}^{2} \left( x - \frac{1}{x^2} \right) \, dx \].
03
Integrate the expression
To find the area, integrate the expression \(x - \frac{1}{x^2}\) with respect to \(x\):\[ \int \left( x - \frac{1}{x^2} \right) \, dx = \int x \, dx - \int \frac{1}{x^2} \, dx \].
04
Solve the individual integrals
Calculate each integral separately:- \(\int x \, dx = \frac{x^2}{2} + C\)- \(\int \frac{1}{x^2} \, dx = \int x^{-2} \, dx = -x^{-1} + C = -\frac{1}{x} + C\).
05
Evaluate the definite integral
Evaluate the definite integral from 1 to 2:\[ A = \left[ \frac{x^2}{2} + \frac{1}{x} \right]_{1}^{2} \].Calculate:\[ = \left( \frac{2^2}{2} + \frac{1}{2} \right) - \left( \frac{1^2}{2} - 1 \right) \].
06
Compute the final result
Calculate the numerical result:- Upper bound (at \(x = 2\)): \(\frac{4}{2} + \frac{1}{2} = 2 + 0.5 = 2.5\)- Lower bound (at \(x = 1\)): \(\frac{1}{2} - 1 = 0.5 - 1 = -0.5\)Thus, \(A = 2.5 - (-0.5) = 2.5 + 0.5 = 3\).
07
Conclusion: Final area
The area of the region bounded by these lines and curves in the first quadrant is 3 square units.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
Definite integrals are a fundamental concept in calculus, essential for determining the total accumulation of a quantity. Imagine you want to calculate the area under a curve or between two curves. The definite integral helps you do this succinctly.
When you evaluate a definite integral, you are looking for the accumulated area between a function and the horizontal axis over a closed interval \([a, b]\). It is denoted as \(\int_{a}^{b} f(x) \, dx\), where \([a, b]\) specifies the interval and \(int f(x)\, dx\) is the integrand.
Key properties of definite integrals include:
When you evaluate a definite integral, you are looking for the accumulated area between a function and the horizontal axis over a closed interval \([a, b]\). It is denoted as \(\int_{a}^{b} f(x) \, dx\), where \([a, b]\) specifies the interval and \(int f(x)\, dx\) is the integrand.
Key properties of definite integrals include:
- Additivity: \(\int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx\)
- Reversal of limits changes the sign: \(\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx\)
- Constant multiple rule: \(\int_{a}^{b} kf(x) \, dx = k\int_{a}^{b} f(x) \, dx\)
Area Between Curves
Finding the area between curves is a practical application of definite integrals. This method is key when you have two curves and you want to know the area contained between them over a certain interval.
To find this area, you typically:
To find this area, you typically:
- Identify the two functions: the upper curve \(y=f(x)\) and the lower curve \(y=g(x)\).
- Determine the interval \([a, b]\) over which the curves are intersecting or desired.
- Set up the integral for the area as \(A = \int_{a}^{b} (f(x) - g(x)) \, dx\).
Applications of Integration
Integration is a powerful tool in mathematical analysis, providing solutions to a multitude of problems beyond simple area calculations. The versatility of integration lies in its wide array of applications, from physics to economics.
Some common real-world applications include:
Some common real-world applications include:
- Finding the total distance traveled by an object by integrating velocity over time.
- Calculating the work done by a variable force along a path.
- Determining the center of mass for an object with variable density.
- Evaluating consumer and producer surplus in economics.
