Question

Which of the following gives the length of the path described by the parametric equations x(t)=2+3tandy(t)=1+tfromt=0tot=1?a. ∫10√1+4t29dtb. ∫10√1+4t2dtc. ∫10√3+3t+t2dtd. ∫10√9+4t2dte. ∫10√(2+3t)2)+(1+t2)2dt

Answer 1

The correct integral for the arc length of the path described by the given parametric equations is option b, the integral from 0 to 1 of the square root of 10 dt.

Step-by-step explanation:

The length of the path described by the parametric equations x(t) = 2 + 3t and y(t) = 1 + t from t = 0 to t = 1 is given by the integral that calculates the arc length of a parametric curve. To find this, we use the arc length formula which involves taking the square root of the sum of the squares of the derivatives of x(t) and y(t) with respect to t and integrating that expression with respect to t over the given interval.

The derivatives are dx/dt = 3 and dy/dt = 1. Substituting these into the arc length formula, we get √(3² + 1²) = √(9 + 1) = √10. The integral for the arc length is therefore ∫ dx/dt √(dx/dt² + dy/dt²) dt = ∫√10 dt, which is evaluated from t = 0 to t = 1.

The answer is option b. ∫01√(9 + 1) dt = ∫01√10 dt, which simplifies to √10 [b].

Answer 2

The correct integral to represent the length of the path described by the parametric equations is:

`L=\int^1_0 \sqrt (1 + 4t^2\ dt)` Option B

How to calculate the length of the path

Let's calculate the length of the path described by the parametric equations x(t)=2+3t and y(t)=1+t from t=0 to t=1.

The formula for the length of a parametric curve from t=a to t=b is given by:

`L=\int^a_b \sqrt ((dx/dt)^2 + (dy/dt)^2 dt)`

Here,

x(t)=2+3t and y(t)=1+t. Let's find dx/dt and dy/dt:

`dx/dt = dx/dt = d(2 + 3t)/dt = 3\\dy/dt = dy/dt = d(1 + t)/dt = 1`

Now, substitute these derivatives into the formula for the length:

`L=\int^1_0 \sqrt ((3)^2 + (1)^2 dt)`

`L=\int^1_0 \sqrt (9 + 1 dt)`

`L=\int^1_0 \sqrt (10\ dt)`

The correct integral to represent the length of the path described by the parametric equations is:

`L=\int^1_0 \sqrt (1 + 4t^2\ dt)`