Calculate Exact Area Calculation by taking limits & find total distance traveled from a velocity vs time Graph.

Consider a general region bounded by \(y=f(x)\) in the interval \([a,b]\) .

- Divide the length \((b-a)\) into \(n\) equal parts, each equal to \(\Delta x = \dfrac{b-a}{n}\)

\(x_1 =a+\Delta x\)

\(x_2 = a+2\,\Delta\,x\)

\(\vdots\)

\(x_{n-1}\;=a+(n-1)\,\Delta \,x\)

As \(n\) increases these rectangles become thinner and thinner and approximation become more and more accurate.

- When \(x \to\infty\) this becomes exact area.

\(\therefore A= \lim\limits _{x\to\infty} R_n = \lim\limits _{x\to\infty} L_n \) both limits will take the same value .

\(A= \lim\limits _{x\to\infty} R_n\, =\lim\limits _{x\to\infty}\underbrace {\left(\dfrac{b-a}{n}\right)}_{\text {Width of each rectangle}}\left[\underbrace {f(x_1)}_{\text {Height of 1st rectangle}}+f(x_2)+.....f(x_n)\right]\)

\(=\lim\limits_{x\to\infty}(\Delta\,x)\,\sum\limits_{i=1}^n\,f(x_i) \to\Delta\,x= \dfrac{b-a}{n}, \,x_i = a+i\,\Delta\,x \ and \;\sum\) means the value of \(i\)varies from 1 to \(n \) .

\(A=\lim\limits_{x\to\infty}L_n=\lim\limits_{x\to\infty}\left( \dfrac{b-a}{n}\right)[f(a)+f(x_1)+.....f(x_{n-1})]\)

\(= \lim\limits_{x\to\infty}\,(\Delta\,x)\sum\limits^n_{i=1} f(x_{i-1})\to (a=x_0)\)

\(\Rightarrow A=\lim\limits_{x\to\infty}\,\sum\limits_{i=1}^n f(x_{i}) \,\Delta\,x = \lim\limits_{x\to\infty} \,\,\sum \limits ^n_{i=1} f(x_{i-1})\,\Delta \,x\)

\(\therefore\) Area = \(\lim\limits_{n\to \infty}\,\sum\limits_{i=1}^{n}\, \Big ( f(a+i\,\Delta x) \;\Delta x\Big) \)

Area bounded by \(y=f(x)\) and the lines \(x=a\) and \(x=b\), where \(\Delta x=\dfrac {b-a}{n}\).

A \(\lim\limits_{n\to\infty}\,\sum\limits_{i=1}^n\dfrac{1}{n} \sqrt{2+i\,n}\)

B \(\lim\limits_{n\to\infty}\,\sum\limits_{i=1}^n\,\dfrac{4}{n} \sqrt{9+\dfrac{28\,i\,}{n}}\)

C \(\lim\limits_{n\to\infty}\,\sum\limits_{i=1}^n\,\dfrac{3}{n} \sqrt{2+\dfrac{3\,i\,}{n}}\)

D \(\lim\limits_{n\to\infty}\,\sum\limits_{i=1}^n\,\left(\dfrac{7}{n} \sqrt{1+\dfrac{\,i\,}{n}}\right)\)

A Region bounded by \(y=x^4\) between \(x=7\) and \(x=11\)

B Region bounded by \(y=x^2\) between \(x=2\) and \(x=6\)

C Region bounded by \(y=sin\,x\) between \(x=\dfrac {\pi}{4}\) and \(x=\dfrac {\pi}{2}\)

D Region bounded by \(y=cos\,x\) between \(x=0\) and \(x=\pi\)

- Consider the velocity time graph of a particle as shown.

- If we break the time interval [a, b] into many equal parts (say n) each measuring \(\dfrac {b-a}{n}=\Delta t\), and draw rectangles similar to area problem, we see that area of each rectangle will be the distance covered by particle in that small time interval..
- Let \(v=f(t)\), \((a\leq t \leq b)\) \([\,f(t)>0\,]\) .
- Distance = \(f(t_0)\Delta t+f(t_1)\Delta t+\dots f(t_{n-1})\Delta t\)\(=\sum\limits_{i=1}^{n-1}\,f(t_{i-1})\Delta t\) (If we take the left end point for height of rectangle)
- Distance = \(f(t_1)\Delta t+f(t_2)\Delta t+\dots f(t_{n})\Delta t\)\(=\sum\limits_{i=1}^{n}\,f(t_{i})\Delta t\) (If we take the right end point for height of rectangle)

Instead of using Left end points \((L_n)\) or Right end points \((R_n)\) for approximating area,

we can use height of any point \(x_i^*\) which lies between \(x_{i-1}\) and \(x_i\).

\(x_1^*\to\) between \(x_0\) and \(x_1\) (or \(a\) and \(x_1\))

\(x_2^*\to\) between \(x_1\) and \(x_2\)

\(\vdots\)

\(x_n^*\to\) between \(x_{n-1}\) and \(b\)

- Therefore, general expression for area \(A=\lim\limits_{x\to\infty}\;f(x_1^*)\,\Delta x+ f(x_2^*)\,\Delta x+\dots f(x_n^*)\,\Delta x\)

\(=\lim\limits_{x\to\infty}\; \sum\limits_{i=1}^n\,f(x_i^*) \Delta x\)

- If we choose \(x_i^*\) to be the maximum height in each respective sub interval, we get the upper sum or the upper estimate of the area.
- If we choose \(x_i^*\) to be the minimum height in each respective sub interval we get the lower sum or the lower estimate of the area.

Tall rectangles \(\to\)upper sum.

A \(x_5^*=6.2\)

B \(x_1^*=5.8\)

C \(x_2^*=9\)

D \(x_5^*=4\)

For constant velocity motion, the distance traveled by a particle is given by

Distance = Speed × time

D = v × t

- If 'v' varies with time, then estimate the distance by taking suitable sub intervals of time and assuming that velocity is constant in these intervals.