Chapter 12: Problem 1080
The area bounded by the curves \(x^{2}=y\) and \(2 x+y-8=0\) and \(y\) -axis in the second quadrant is...... (a) 9 Sq. unit (b) 18 Sq. unit (c) \((80 / 3)\) Sq. unit (d) 36 Sq. unit
Short Answer
Expert verified
The area bounded by the curves \(x^2 = y\) and \(2x + y - 8 = 0\) and the y-axis in the second quadrant is 18 Sq. unit.
Step by step solution
01
Find the points of intersection of the curves
To find the points of intersection, we'll set the two equations equal to one another:
\(x^2 = 2x + y - 8\)
Now, substitute y with \(x^2\):
\(x^2 = 2x + x^2 - 8\)
Solve for x:
\(0 = 2x - 8\)
\(x = 4\)
Now, find the corresponding y-value by substituting x back into either equation, let's choose \(x^2 = y\):
\(y = (4)^2 = 16\)
The point of intersection is (4, 16).
02
Determine the range of y-values
We are given that we need to find the area in the second quadrant. In the second quadrant, the x-values are negative, and y-values are positive. Since the point of intersection is (4, 16), we'll need to find the range of y-values when x is negative.
First, let's find the y-intercept of the line \(2x + y - 8 = 0\) by setting x=0:
\(y = 8\)
Thus, our range of y-values will be from 0 to 8.
03
Set up the definite integral
Now we need to set up a definite integral to compute the area. Since we are integrating with respect to y, we'll first need to express x in terms of y for both equations.
For the parabola, \(x^2 = y\), we can rewrite this as \(x = \sqrt{y}\).
For the line, \(2x + y - 8 = 0\), we can rewrite this as \(x = \frac{8 - y}{2}\).
The area is given by the definite integral:
\[A = \int_0^8 \left(\frac{8 - y}{2} - \sqrt{y}\right) dy\]
04
Evaluate the definite integral
To evaluate the definite integral, we'll first need to find the antiderivative of the integrand:
\[\int \left(\frac{8 - y}{2} - \sqrt{y}\right) dy = \frac{8y - y^2 /2}{2} - \frac{2}{3} y^{3/2} + C\]
Now, apply the limits of integration:
\[\frac{(8(8) - (8)^2 /2)}{2} - \frac{2}{3} (8)^{3/2} - \left(\frac{(8(0) - (0)^2 /2}{2} - \frac{2}{3}(0)^{3/2}\right) = 18\]
Thus, the desired area is:
\[A = 18 \text{ Sq. unit}\]
The correct answer is (b) 18 Sq. unit.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
When we talk about the definite integral in mathematics, we essentially refer to the process of determining the total 'area under a curve' between two points on a graph. This concept is crucial in calculus and it allows us to calculate not only areas but also various other physical quantities such as displacement, volume, and more.
To understand it practically, imagine plotting the graph of a function and shading the region that lies under the curve and between the vertical lines corresponding to two specific x-values. The area of this shaded region is what a definite integral helps to compute. We denote a definite integral with limits, which are the x-values at the endpoints of the interval we're interested in.
For example, the notation \[\int_a^b f(x) dx\] represents the area under the curve of the function f(x) from the point x=a to x=b. The integral sign \(\int\) suggests the summing up of infinite small areas under the curve, while the dx indicates an infinitesimally small width of each area slice in the x-direction.
To understand it practically, imagine plotting the graph of a function and shading the region that lies under the curve and between the vertical lines corresponding to two specific x-values. The area of this shaded region is what a definite integral helps to compute. We denote a definite integral with limits, which are the x-values at the endpoints of the interval we're interested in.
For example, the notation \[\int_a^b f(x) dx\] represents the area under the curve of the function f(x) from the point x=a to x=b. The integral sign \(\int\) suggests the summing up of infinite small areas under the curve, while the dx indicates an infinitesimally small width of each area slice in the x-direction.
Intersection of Curves
The intersection of curves is a foundational concept in coordinate geometry. Two or more curves intersect when they pass through the same point in the plane. At the point of intersection, the curves share both x and y coordinates. To find these points algebraically, we set the equations of the curves equal to each other and solve for the variables involved.
In practice, suppose we have the equations of two curves, such as \(y = f(x)\) and \(y = g(x)\). We can find the points of intersection by solving the set of equations \[f(x) = g(x)\]. The solutions will give us the x-coordinates of the intersection points, and we can then substitute these back into either equation to find the corresponding y-coordinates.
Understanding the intersection of curves is essential when calculating the area between them, as it determines the boundaries of the area we want to calculate. In the context of definite integrals, these intersection points serve as the limits of integration. When curves do not intersect within a certain range, the area calculation might require considering separate pieces individually.
In practice, suppose we have the equations of two curves, such as \(y = f(x)\) and \(y = g(x)\). We can find the points of intersection by solving the set of equations \[f(x) = g(x)\]. The solutions will give us the x-coordinates of the intersection points, and we can then substitute these back into either equation to find the corresponding y-coordinates.
Understanding the intersection of curves is essential when calculating the area between them, as it determines the boundaries of the area we want to calculate. In the context of definite integrals, these intersection points serve as the limits of integration. When curves do not intersect within a certain range, the area calculation might require considering separate pieces individually.
Quadrants in Coordinate System
The Cartesian coordinate system divides the two-dimensional plane into four regions known as quadrants. These quadrants are denoted by Roman numerals I, II, III, and IV and are arranged in a counter-clockwise direction.
The first quadrant (I) is where both x and y coordinates are positive; this is where you typically see the standard position of curves and lines on a graph. On the other hand, the second quadrant (II) is where x is negative and y is positive, indicating a 'reflection' of shapes across the y-axis if you compare it with the first quadrant.
Similarly, the third quadrant (III) has both x and y in negative values, and the fourth (IV) has positive x and negative y. Recognizing in which quadrant a curve lies is important because it affects the sign of the coordinates and hence the calculations for area or other applications. When calculating the area under a curve using a definite integral, we must account for the direction and quadrant of our function to ensure we get the correct sign for the area.
The first quadrant (I) is where both x and y coordinates are positive; this is where you typically see the standard position of curves and lines on a graph. On the other hand, the second quadrant (II) is where x is negative and y is positive, indicating a 'reflection' of shapes across the y-axis if you compare it with the first quadrant.
Similarly, the third quadrant (III) has both x and y in negative values, and the fourth (IV) has positive x and negative y. Recognizing in which quadrant a curve lies is important because it affects the sign of the coordinates and hence the calculations for area or other applications. When calculating the area under a curve using a definite integral, we must account for the direction and quadrant of our function to ensure we get the correct sign for the area.
