Consider the curve given by [math]e^x = y^3 + 10[/math]a. Set up an integral in terms of x that represents the arclength from x = 1 to x = 2. Then evaluate your integral using a computer system.
b. Set up an integral in terms of y that represents the arclength from x = 1 to x = 2. then evaluate your integral using a computer system and make sure it matches with part a.
My answers:
a. I isolated y to get [math]y = cuberoot(e^x - 10)[/math] Then I took the derivative of that to get e^x / (3(e^x-10)^(2/3). My integrand for the definite integral from 1 to 2 was then sqrt(1 + (e^x / (3(e^x - 10))^2/3)^2). Inputting this into integral-calculator.com, it said that there was no elementary antiderivative but that the approximation was 1.169606600612212. Is this correct?
b. I isolated x to get [math]x = ln(y^3 + 10)[/math] Then I took the derivative of that and got [math]3y^2/(y^3+10)[/math]. Then my integrand was sqrt(1+(3y^2/(y^3 + 10))^2). Inputting this into integral-calculator.com, it said that there was no elementary antiderivative but that the approximation was 1.119682131946315. The two answers are not the same, but since they're approximations, is this correct anyways? Was my process on both of these correct to begin with or am I making a mistake somewhere? Thank you!
b. Set up an integral in terms of y that represents the arclength from x = 1 to x = 2. then evaluate your integral using a computer system and make sure it matches with part a.
My answers:
a. I isolated y to get [math]y = cuberoot(e^x - 10)[/math] Then I took the derivative of that to get e^x / (3(e^x-10)^(2/3). My integrand for the definite integral from 1 to 2 was then sqrt(1 + (e^x / (3(e^x - 10))^2/3)^2). Inputting this into integral-calculator.com, it said that there was no elementary antiderivative but that the approximation was 1.169606600612212. Is this correct?
b. I isolated x to get [math]x = ln(y^3 + 10)[/math] Then I took the derivative of that and got [math]3y^2/(y^3+10)[/math]. Then my integrand was sqrt(1+(3y^2/(y^3 + 10))^2). Inputting this into integral-calculator.com, it said that there was no elementary antiderivative but that the approximation was 1.119682131946315. The two answers are not the same, but since they're approximations, is this correct anyways? Was my process on both of these correct to begin with or am I making a mistake somewhere? Thank you!
