Let F(x)=∫x^x^2+(π)/(6) 2 cos^2 t d t for all x ∈R and f:[0,...

Let $F(x)=\int_{x}^{x^{2}+\frac{\pi}{6}} 2 \cos ^{2} t d t$ for all $x \in R$ and $f:\left[0, \frac{1}{2}\right] \rightarrow[0, \infty)$ be a continuous function. For $a \in\left[0, \frac{1}{2}\right]$, if $F^{\prime}(a)+2$ is the area of the region bounded by $x=0, y=0, y=f(x)$ and $x=a$, then $f(0)$ is

Let $F(x)=\int_{x}^{x^{2}+\frac{\pi}{6}} 2 \cos ^{2} t d t$ for all $x \in R$ and
$f:\left[0, \frac{1}{2}\right] \rightarrow[0, \infty)$ be a continuous function. For $a \in\left[0, \frac{1}{2}\right]$, if $F^{\prime}(a)+2$ is the area of the region bounded by $x=0, y=0, y=f(x)$ and $x=a$, then $f(0)$ is

Key Concepts

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, indicating that the process of integration can be reversed by differentiation. It ensures that if a function is continuous, then its definite integral can be differentiated to recover the original integrand, a property crucial for evaluating and understanding integrals with variable limits.

Leibniz Integral Rule

The Leibniz Integral Rule generalizes the Fundamental Theorem of Calculus to situations where the limits of integration are functions of the variable. It provides a way to differentiate an integral whose limits are not constant by accounting for both the variation of the integrand and the change in the limits through their derivatives.

Chain Rule

The chain rule is a basic differentiation technique used for composite functions. When applying the Leibniz Integral Rule, the chain rule is essential because the upper and/or lower limits of integration are often functions of the variable, requiring the multiplication by the derivatives of these limits.

Interpretation of Definite Integrals as Areas

Definite integrals can be interpreted as the area under a curve within specified limits, which is a fundamental concept in calculus. This interpretation is particularly important when relating the value of an integral to geometric quantities such as the region bounded by specific curves or lines.

Continuity of Functions

Ensuring that functions are continuous is critical in calculus, especially when applying the Fundamental Theorem of Calculus and differentiation under the integral sign. Continuity guarantees that integrals and derivatives behave in a predictable manner, making it possible to equate areas with integrals and to differentiate integrals with variable limits.

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