Q. 4 - Calculus

Question

State the Fundamental Theorem of Calculus 

(a) in its original form, 

(b) in its alternative form, and 

(c) by using an indefinite integral and evaluation notation.

Step-by-Step Solution

Answer

(a) The fundamental Theorem of calculus in its original form is ${\int }_a^{b}f\left(x\right).dx=F\left(b\right)-F\left(a\right)$.

(b) The fundamental Theorem of calculus in its alternative form is ${\left[F\right(x\left)\right]}_a^b=F\left(b\right)-F\left(a\right)$.

(c) The fundamental Theorem of calculus in its by using an indefinite integral and evaluation notation is ${\int }_a^{b}f\left(x\right).dx={[\int f(x).dx]}_a^b$.

Part (a) Step 1. Given Information.

The fundamental theorem of calculus.

Part (a) Step 2. Original form.

If f is continuous on [a, b] and F is any antiderivative of f , then

${\int }_a^{b}f\left(x\right).dx=F\left(b\right)-F\left(a\right)$

Part (b) Step 1. Alternative form.

For any function F on an interval [a, b], the difference F(b)−F(a) will be called the evaluation of F(x) on [a, b] and will be denoted by ${\left[F\right(x\left)\right]}_a^b=F\left(b\right)-F\left(a\right)$.

Part (c) Step 1. Evaluation Notation.

If f is continuous on [a, b], then

${\int }_a^{b}f\left(x\right).dx={[\int f(x).dx]}_a^b$.