According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value ξ in this interval such that \(\mathop \smallint \limits_a^b f\left( x \right)dx =\)
f (ξ) (b - a)
Mean Value Theorem for Integrals Explained
The Mean Value Theorem is a key concept in calculus that connects the average value of a function over an interval to a specific point within that interval. When applied to integrals, it helps us understand the relationship between the total area under a curve and the function's value at a particular point.
Continuous Function and Theorem Statement
For a function \(f(x)\) to satisfy the conditions of the Mean Value Theorem for Integrals, it must be continuous over a closed interval \([a, b]\). If this condition is met, the theorem guarantees that there exists at least one value, denoted as \(\xi\) (pronounced "xi"), within the open interval \((a, b)\). This special value \(\xi\) has the property that the definite integral of \(f(x)\) from \(a\) to \(b\) is equal to the product of the function's value at \(\xi\) and the length of the interval \((b - a)\).
In simpler terms, you can find a point \(\xi\) in the interval such that the rectangle formed by the height \(f(\xi)\) and the width \((b-a)\) has an area exactly equal to the area under the curve of \(f(x)\) from \(a\) to \(b\).
Integral Formula Derivation
The mathematical statement of the Mean Value Theorem for Integrals is given by the following formula:
\(\mathop \smallint \limits_a^b f\left( x \right)dx = f(\xi)(b-a)\)
Let's break down the components of this formula:
- \(\mathop \smallint \limits_a^b f\left( x \right)dx\): This represents the definite integral of the function \(f(x)\) over the interval \([a, b]\). It calculates the net signed area between the graph of \(f(x)\) and the x-axis from \(x=a\) to \(x=b\).
- \(f(\xi)\): This is the value of the function \(f\) at the specific point \(\xi\). This value is often referred to as the "average value" of the function over the interval \([a, b]\).
- \((b - a)\): This represents the length or width of the interval \([a, b]\).
So, the theorem essentially states that the total area under the curve is equal to the average height of the function multiplied by the width of the interval.
Option Analysis and Correct Form
Now, let's compare the established formula from the Mean Value Theorem for Integrals with the given options to identify the correct one:
| Option | Expression | Analysis based on Theorem |
|---|---|---|
| 1 | \(f (\xi) (b - a)\) | This expression directly matches the formal statement of the Mean Value Theorem for Integrals, where the integral is equal to the function value at \(\xi\) multiplied by the length of the interval. |
| 2 | \(f (b) (\xi - a)\) | This expression is incorrect because the function should be evaluated at \(\xi\), and the interval length should be \((b-a)\), not \((\xi-a)\). |
| 3 | \(f (a) (b - \xi)\) | This expression is also incorrect. The function should be evaluated at \(\xi\), and the length of the interval is \((b-a)\), not \((b-\xi)\). |
| 4 | \(0\) | This option suggests the integral is always zero, which is generally false. The integral is zero only in very specific cases (e.g., if the function itself is zero over the interval, or if positive and negative areas perfectly cancel out). It does not represent the general statement of the Mean Value Theorem. |
Therefore, based on the definition and application of the Mean Value Theorem for Integrals, the integral \(\mathop \smallint \limits_a^b f\left( x \right)dx\) is correctly represented by \(f(\xi)(b-a)\).
Important Questions from Mean Value Theorem
A series expansion for the function sin θ is
If f is the derivative of some function on [a, b], then there exists a number c in (a, b) such that Integral of f with respect to x =
Which condition is not required in checking for Taylor's theorem?
- What is the interval of Taylor series expansion of tan(x)?
Let f(x) = x2 - 2x + 2 be a continuous function defined on x ∈ [1, 3]. The point x at which the tangent of f(x) becomes parallel to the straight line joining f(1) and f(3) is
