How do you find the average distance from the origin of a point on the parabola #y=x^2, 0

Question

How do you find the average distance from the origin of a point on the parabola #y=x^2, 0<=x<=4# with respect to x?

Paisley Johnson · Accepted Answer

#(17sqrt17-1)/12~~5.84# Let #P# be a point on the parabola. The coordinates with respect to #x# will be #(x,x^2)#. We can make a function out of the distance to the origin using the distance formula: #d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)# Applying this to our point #P=(x,x^2)# and the origin #(0,0)#, we get: #d(x)=sqrt((x-0)^2+(x^2-0)^2)# #d(x)=sqrt(x^2+x^4)# To work out the average value of the function, we can use the average value formula. For a function #f(x)# on the interval #[a,b]#, the average value will be: #(int_a^bf(x)\ dx)/(b-a)# In our case, we get: #(int_0^4d(x)\ dx)/(4-0)=1/4int_0^4sqrt(x^2+x^4)\ dx# I will first work out the antiderivative: #int\ sqrt(x^2+x^4)\ dx=# #=int\ sqrt(x^2(1+x^2)\ dx=# #=int\ xsqrt(x^2+1)\ dx=# Now we can introduce a u-substitution with #u=x^2+1#. The derivative of #u# is then #2x#, so we divide through by that: #=int\ (cancel(x)sqrtu)/(2cancel(x))\ du=# #1/2int\ sqrtu\ du=1/2*2/3u^(3/2)+C=2/6(1+x^2)^(3/2)+C# Now we can evaluate the original expression: #1/4int_0^4sqrt(x^4+x^2)\ dx=1/4[1/3(1+x^2)^(3/2)]_0^4=# #=1/4(1/3(1+16)^(3/2)-1/3(1+0)^(3/2))=# #=1/4(1/3*17^(3/2)-1/3*1^(3/2))=1/4((17sqrt(17))/3-1/3)=# #=1/4*(17sqrt17-1)/3=(17sqrt17-1)/12# So, the average distance from a point to the origin on the parabola on the interval #[0,4]# is: #(17sqrt17-1)/12~~5.84#

Emily Ammerman · Answer

undefined To find the average distance from the origin of a point on the parabola $ y = x^2 $, $ 0 \leq x \leq 4 $ with respect to $ x $, you integrate the square root of the sum of squares of $ x $ and $ x^2 $ over the interval $ [0, 4] $, and then divide by the length of the interval $ 4 - 0 = 4 $. This can be expressed by the following formula:

$$ \text{Average distance} = \frac{1}{4} \int_{0}^{4} \sqrt{x^2 + (x^2)^2} \, dx $$

Solving this integral will give you the average distance from the origin.

How do you find the average distance from the origin of a point on the parabola #y=x^2, 0&lt;=x&lt;=4# with respect to x?

How do you find the average distance from the origin of a point on the parabola #y=x^2, 0<=x<=4# with respect to x?