Question
Let $$\left( {a,\,b} \right)$$ and $$\left( {\lambda ,\,\mu } \right)$$ be two points on the curve $$y = f\left( x \right).$$ If the slope of the tangent to the curve at $$\left( {x,\,y} \right)$$ be $$\phi \left( x \right)$$ then $$\int_a^\lambda {\phi \left( x \right)} \,dx$$ is :
A.
$$\lambda - a$$
B.
$$\mu - b$$
C.
$$\lambda + \mu - a - b$$
D.
none of these
Answer :
$$\mu - b$$
Solution :
$$\eqalign{
& {\text{Here}}\,\,f'\left( x \right) = \phi \left( x \right) \cr
& {\text{So}},\,\,\int_a^\lambda {\phi \left( x \right)dx = \int_a^\lambda {f'\left( x \right)dx} } = \left[ {f\left( x \right)} \right]_a^\lambda = f\left( \lambda \right) - f\left( a \right) \cr
& {\text{But}}\,\,b = f\left( a \right),\,\,\mu = f\left( \lambda \right).\,\,{\text{So}},\,\,\int_a^\lambda {\phi \left( x \right)dx = \mu - } b \cr} $$