Problem 33
Question
In Exercises 29-36, complete the table using the function \( f(x) \), over the specified interval [a, b], to approximate the area of the region bounded by the graph of the \( y = f(x) \), the x-axis, and the vertical lines \(x=a\) and \(x=b\) and using the indicated number of rectangles. Then find the exact area as \( n \to \infty \). $$ f(x) = 9 - x^2 $$ Interval \( [0, 2] \)
Step-by-Step Solution
Verified Answer
The exact area under the curve \( y = f(x) = 9 - x^2 \) on the interval [0,2] is 14 square units.
1Step 1: Function Evaluation
The function \(f(x)\) in the exercise is a parabolic function given by \(f(x) = 9 - x^2\). The function needs to be evaluated over the interval [0, 2]. So, for any x-value in this interval, the function \(f(x)\) gives the height of the corresponding rectangle to approximate the area under the curve.
2Step 2: Rectangle Approximation
To approximate the area using rectangles, one must partition the interval [0, 2] into 'n' equal parts (which will be the widths of the rectangles), calculate the height of each rectangle (which is given by \(f(x)\)), and then sum the areas (width x height) of all these rectangles. If 'n' is a large number, this approximation gets closer to the actual area under the curve.
3Step 3: Exact Area Calculation
The exact area under the curve can be obtained by taking the limit as 'n' goes to infinity of the sum of the areas of these rectangles. In other words, it's the integral of the function \(f(x)\) over the interval [0, 2]. To find the exact area, we have to integrate the function \(f(x)=9-x^2\) from 0 to 2. This implies the use of the Fundamental Theorem of Calculus and basic knowledge in taking definite integrals.
Key Concepts
Parabolic FunctionRectangle Approximation MethodDefinite IntegralFundamental Theorem of Calculus
Parabolic Function
A parabolic function is a second-degree polynomial function that can be denoted generally as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero. The graph of this function is a parabola which opens upwards if \( a > 0 \) and downwards if \( a < 0 \). In our exercise, \( f(x) = 9 - x^2 \), the parabola opens downwards since the coefficient of \( x^2 \) is negative.
The shape of the parabola is important when considering the area under the curve between two points on the x-axis. Because the curve is symmetrical, the area on one side of the vertex is a mirror image of the other. This symmetric property can simplify calculations, both for approximation methods and exact integrations, by providing a predictable pattern of the values that the function outputs.
The shape of the parabola is important when considering the area under the curve between two points on the x-axis. Because the curve is symmetrical, the area on one side of the vertex is a mirror image of the other. This symmetric property can simplify calculations, both for approximation methods and exact integrations, by providing a predictable pattern of the values that the function outputs.
Rectangle Approximation Method
The rectangle approximation method is a technique used to estimate the area under a curve. It involves dividing the interval of interest, in this case, \( [0, 2] \), into a number of rectangles (subintervals), calculating the area of each rectangle, and summing them up to get the total approximate area.
The widths of the rectangles are determined by the number of subdivisions \( n \), where the interval length is divided by \( n \). The rectangle heights are the function values at specific points of the subdivision, which commonly include the left end, right end, midpoints, or a combination of these points of the subintervals. \( f(x) \) gives the height of the rectangles. The finer the subdivision (larger \( n \)), the closer the approximation will be to the true area under the curve. This method converges to an exact answer as \( n \to \infty \).
The widths of the rectangles are determined by the number of subdivisions \( n \), where the interval length is divided by \( n \). The rectangle heights are the function values at specific points of the subdivision, which commonly include the left end, right end, midpoints, or a combination of these points of the subintervals. \( f(x) \) gives the height of the rectangles. The finer the subdivision (larger \( n \)), the closer the approximation will be to the true area under the curve. This method converges to an exact answer as \( n \to \infty \).
Definite Integral
A definite integral is a concept that gives the exact area under a curve on a closed interval [a, b]. When computing a definite integral, we are essentially adding up an infinite number of infinitely small quantities along the interval — these quantities are the products of function values and infinitesimally small widths.
The definite integral is denoted as \( \int_{a}^{b} f(x) \, dx \), which reads as the integral of \( f(x) \) with respect to \( x \) from \( a \) to \( b \). Returning to our exercise, to find the exact area under the parabolic curve \( f(x) = 9 - x^2 \) from \( x = 0 \) to \( x = 2 \), we would evaluate the definite integral of \( f(x) \) over this interval. The process of integrating takes into account the precise changes in the height of the curve over the interval, giving the exact area beneath it.
The definite integral is denoted as \( \int_{a}^{b} f(x) \, dx \), which reads as the integral of \( f(x) \) with respect to \( x \) from \( a \) to \( b \). Returning to our exercise, to find the exact area under the parabolic curve \( f(x) = 9 - x^2 \) from \( x = 0 \) to \( x = 2 \), we would evaluate the definite integral of \( f(x) \) over this interval. The process of integrating takes into account the precise changes in the height of the curve over the interval, giving the exact area beneath it.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the concept of differentiation with integration and provides a method for evaluating definite integrals. It states that if a function is continuous on the interval [a, b], and F is an antiderivative of f on [a, b], then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).
This theorem is critical for finding the exact area under a curve, as it allows us to calculate integrals without having to sum infinitesimal quantities directly. In the context of our parabolic function example, we find the antiderivative of \( f(x) = 9 - x^2 \), and evaluate it at the endpoints 0 and 2 to get the exact area. The beauty of the Fundamental Theorem of Calculus is that it turns a potentially laborious and impractical task into a manageable one, provided we know how to find the antiderivative of our function.
This theorem is critical for finding the exact area under a curve, as it allows us to calculate integrals without having to sum infinitesimal quantities directly. In the context of our parabolic function example, we find the antiderivative of \( f(x) = 9 - x^2 \), and evaluate it at the endpoints 0 and 2 to get the exact area. The beauty of the Fundamental Theorem of Calculus is that it turns a potentially laborious and impractical task into a manageable one, provided we know how to find the antiderivative of our function.
Other exercises in this chapter
Problem 32
In Exercises 29-42, find the derivative of the function. \(g(x) = -5x+2\)
View solution Problem 32
In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 0} \dfrac{\dfrac{1}{2+x}-\dfrac{1}{2}}
View solution Problem 33
In Exercises 29-34, use a graphing utility to graph the function and verify that the horizontal asymptote corresponds to the limit at infinity. $$ y = 1-\dfrac{
View solution Problem 33
In Exercises 29-42, find the derivative of the function. \(f(x) = 4-3x^2\)
View solution