(P3T9Q) Arc Length [QUIZ]

(P3T9Q) Arc Length [QUIZ]

• Multi-step

Multiple Choice

STEP 1:

What's the correct formula to calculate Arc Length?

Reorder

STEP 2:

Calculate the derivative of the function $$\ln\left(cos\left(x\right)\right)$$ .

1.-Raw derivative.

2.-Simplified derivative.

3.-Substitution using a trig identity.

• $$\frac{1}{\cos\left(x\right)}\left(-\sin\left(x\right)\right)$$

• $$-\frac{\sin\left(x\right)}{\cos\left(x\right)}$$

• $$-\tan\left(x\right)$$

1

2

3

• 1
• 2
• 3

Drag and Drop

$$\int0^{\frac{\pi}{4}}\sqrt[]{1+\left[-\tan\left(x\right)\right]^2}dx$$

STEP 3:

After substituting everything in the formula.

Simplify the expression $$\left[-\tan\left(x\right)\right]^2$$ .

​

Placeholder

Drag these tiles and drop them in the correct blank above

Drag and Drop

$$\int0^{\frac{\pi}{4}}\sqrt[]{1+\tan^2\left(x\right)}dx$$

STEP 4:

Keep on simplifying.

Google "trig identities" and choose an equivalent expression for

$$1+\tan^2\left(x\right)$$ :

​ ​

Placeholder

Drag these tiles and drop them in the correct blank above

Drag and Drop

$$\int_0^{\frac{\pi}{4}}\sqrt[]{\sec^2\left(x\right)}dx$$

STEP 5:

Keep on simplifying.

Choose an equivalent expression for

$$\sqrt[]{\sec^2\left(x\right)}$$ :

​ ​ ​

Placeholder

Drag these tiles and drop them in the correct blank above

Multiple Choice

$$\int_0^{\frac{\pi}{4}}\sec\left(x\right)dx$$

STEP 6:

Look in your notebook for the antiderivative of $$sec\left(x\right)$$ .

• $$\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|$$

• $$-\ln\left|\csc\left(x\right)+\cot\left(x\right)\right|$$

• $$\tan\left(x\right)$$

• $$-\cot\left(x\right)$$

Math Response

$$\left[\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|\right]_0^{\frac{\pi}{4}}$$

STEP 7:

Evaluate the antiderivative at the upper bound.

(write down the answer with three decimals without rounding)

USE RADIANS!

Type answer here

Correct Answer: 0.881

Deg^°

Rad

Math Response

$$\left[\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|\right]_0^{\frac{\pi}{4}}$$

STEP 8:

Evaluate the antiderivative at the lower bound.

(write down the answer with three decimals without rounding)

USE RADIANS!

Type answer here

Correct Answer: 0

Deg^°

Rad

Math Response

$$\left[\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|\right]_0^{\frac{\pi}{4}}$$

STEP 9:

FINAL ANWER

What you got from the upper minus that of the lower.

(write down the answer with three decimals without rounding)

USE RADIANS!

Type answer here

Correct Answer: 0.881

Deg^°

Rad

• Multi-step

Math Response

STEP 1:

Calculate the derivative of the function $$x^2-\frac{ln\left(x\right)}{8}$$ .

Type answer here

Correct Answer: 2x-\frac{1}{8x}

Deg^°

Rad

Math Response

$$\int_1^{10}\sqrt[]{1+\left[2x-\frac{1}{8x}\right]^2}dx$$

STEP 2:

After substituting everything in the formula.

Simplify the expression $$\left[2x-\frac{1}{8x}\right]^2$$ .

Use binomial squared:

$$\left(a+b\right)^2=a^2+2ab+b^2$$

Type answer here

Correct Answer: 4x^2-\frac{1}{2}+\frac{1}{64x^2}

Deg^°

Rad

Math Response

$$\int_1^{10}\sqrt[]{1+4x^2-\frac{1}{2}+\frac{1}{64x^2}}dx$$

STEP 3:

Keep on simplifying.

Simplify the expression $$1+4x^2-\frac{1}{2}+\frac{1}{64x^2}$$ .

Type answer here

Correct Answer: 4x^2+\frac{1}{2}+\frac{1}{64x^2}

Deg^°

Rad

Math Response

$$\int_1^{10}\sqrt[]{4x^2+\frac{1}{2}+\frac{1}{64x^2}}dx$$

STEP 4:

Keep on simplifying.

Factorize $$4x^2+\frac{1}{2}+\frac{1}{64x^2}$$ .

Use binomial squared:

$$\left(a+b\right)^2=a^2+2ab+b^2$$

Type answer here

Correct Answer: \left(2x+\frac{1}{8x}\right)^2

Deg^°

Rad

Math Response

$$\int_1^{10}\sqrt[]{\left(2x+\frac{1}{8x}\right)^2}dx$$

STEP 5:

Keep on simplifying.

Simplify $$\sqrt[]{\left(2x+\frac{1}{8x}\right)^2}$$ .

Type answer here

Correct Answer: 2x+\frac{1}{8x}

Deg^°

Rad

Math Response

$$\int_1^{10}2x+\frac{1}{8x}dx$$

STEP 6:

Calculate the antiderivative of $$2x+\frac{1}{8x}$$ .

Type answer here

Correct Answer: x^2+\frac{1}{8}\ln\left(x\right)

Deg^°

Rad

Math Response

$$\left[x^2+\frac{1}{8}\ln\left(x\right)\right]_1^{10}$$

STEP 7:

Evaluate the antiderivative at the upper bound.

(write down the answer with three decimals without rounding)

Type answer here

Correct Answer: 100.287

Deg^°

Rad

Math Response

$$\left[x^2+\frac{1}{8}\ln\left(x\right)\right]_1^{10}$$

STEP 8:

Evaluate the antiderivative at the lower bound.

(write down the answer with three decimals without rounding)

Type answer here

Correct Answer: 1

Deg^°

Rad

Math Response

$$\left[x^2+\frac{1}{8}\ln\left(x\right)\right]_1^{10}$$

STEP 9:

FINAL ANWER

What you got from the upper minus that of the lower.

(write down the answer with three decimals without rounding)

Type answer here

Correct Answer: 99.287

Deg^°

Rad