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 \([a,b]\) .
\(x_1 =a+\Delta x\)
\(x_2 = a+2\,\Delta\,x\)
\(\vdots\)
\(x_{n-1}\;=a+(n-1)\,\Delta \,x\)
As \(n\) increases these rectangles become thinner and thinner and approximation become more and more accurate.
\(\therefore A= \lim\limits _{x\to\infty} R_n = \lim\limits _{x\to\infty} L_n \) 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 \(x=\pi\)
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 \(b\)
\(=\lim\limits_{x\to\infty}\; \sum\limits_{i=1}^n\,f(x_i^*) \Delta x\)
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