# Geometrical Interpretation of definite Integral

## Geometrical Interpretation of definite Integral

If f(x) > 0 for all x ϵ [a, b] then $$\int_{a}^{b}{f\left( x \right)}dx$$ is numerical equal to the area bounded by the curve y = f(x) then x-axis and the straight lines x = a and x = b.

$$\int_{a}^{b}{f\left( x \right)}\left( dx \right)$$ = Area LAMB. In general, $$\int_{a}^{b}{f\left( x \right)}dx$$ represented an algebraic sum of the areas of the figure bounded by the curve y=f(x), the x-axis and the straight lines x = a and x = b. the areas above x-axis are taken with plus sign and the below x-axis are taken with a minus sign.

$$\int_{a}^{b}{f\left( x \right)dx}$$ = Area LAP – Area PQR + Area RBM. Evaluate: $$\int_{a}^{b}{{{e}^{x}}}dx$$ a limit of suns.

Solution: $$\underset{h\to 0}{\mathop{\lim }}\,h\left[ f\left( a \right)+f\left( a+h \right)+f\left( a+2h \right)+….+f\left( a+\left( n-1 \right)h \right) \right]$$ where $$h=\frac{b-a}{n}$$.

Here, f(x) = ex

∴ $$\int_{a}^{b}{{{e}^{x}}}dx$$,

$$=\underset{h\to 0}{\mathop{\lim }}\,h\left[ {{e}^{a}}+{{e}^{a+h}}+{{e}^{a+2h}}+….+{{e}^{a+\left( n-1 \right)h}} \right]$$,

$$=\underset{h\to 0}{\mathop{\lim }}\,h{{e}^{a}}\left[ 1+{{e}^{h}}+{{2}^{2h}}+{{e}^{3h}}+….+{{e}^{\left( n-1 \right)h}} \right]$$,

$$=\underset{h\to 0}{\mathop{\lim }}\,h{{e}^{a}}\left[ \frac{{{\left( {{e}^{h}} \right)}^{n}}-1}{{{e}^{h}}-1} \right]\,\,\,\,\left[ \because a+ar+….+a{{r}^{n-1}}=a\left( \frac{{{r}^{n}}-1}{r-1} \right) \right]$$,

$$=\underset{h\to 0}{\mathop{\lim }}\,h{{e}^{a}}\left( \frac{{{e}^{nh}}-1}{{{e}^{h}}-1} \right)$$,

$$=\underset{h\to 0}{\mathop{\lim }}\,{{e}^{a}}\left[ \frac{\left( {{e}^{b-a}}-1 \right)}{\left( \frac{{{e}^{h}}-1}{h} \right)} \right]={{e}^{a}}\left( {{e}^{b-a}}-1 \right)={{e}^{b}}-{{e}^{a}}$$.