40 DIFFERENTIAL AND INTEGRAL CALCULUS. 



If P n = f x n V(- 2 + 3z - x*} dx, 



J t 



prove that 2 (n + 2) P H - 3 (2n - 1) P w . x + 4 (w - 1) P N _ 2 = 0. 

 8. Obtain the complete areas of the ellipses, 



(1) ax* + 2hxy + by*=l, 

 (2) ax* + by* + c + 2fy + 2gx + 2hxy = 0. 

 For any value of or, we have, in (1), 



whence, if y^ y >2 be these two values of y, the element of 

 the integral is (y z y^dx^ and this must extend over the 

 values of x for which y l and y 2 continue possible ; hence, 

 if we put b = m z (ab A 2 ), the limits of a; are - w, m, and the 



area= 



2 ,, , _ 7T . 2 7T 



2 



(2) treated in exactly the same way, gives the area 



TT (af + If + c/i 2 - a c - 2^A) 4- (a J - A") 1 ' 

 9. Obtain the whole area of the loop of the curve 



fl T* 



/* = a; 8 ; also of the area between the curve and the 



asymptote. (1) 2 (2 - j) , (2) TO* (2 -f | ) . 



DIFFERENTIATION OF A DEFINITE INTEGRAL WITH 



RESPECT TO SOME SYMBOL INVOLVED. 



Suppose u \ (j)(x,z)dx, where z is a quantity in- 



a 



dependent of a?, and also of a and 5, then if 



