Let's analyze each statement step by step: --- ### (a) \(\int_0^{\pi/4}(1-\sqrt{\tan x})dx = \int_0^1 2x \arctan((1-x)^2)dx\) Let's try substitution for the RHS: Let \( u = (1-x)^2 \Rightarrow du = -2(1-x)dx \), but the bounds change from \( x=0 \Rightarrow u=1 \), \( x=1 \Rightarrow u=0 \). But this substitution doesn't immediately match the LHS form. Checking with a calculator or by integration, these two are **not obviously equal**. **Answer:** **False** --- ### (b) \(\int_0^1 2x \frac{1-\sqrt{x}}{\sqrt{x}+4} dx = \int_0^{1/4} \left(\frac{1-4x}{1+x}\right)^2 dx\) Substitute \( x = t^2 \) in the LHS: - When \( x = 0, t = 0 \) - When \( x = 1, t = 1 \) - \( dx = 2t dt \) So, \[ 2x \frac{1-\sqrt{x}}{\sqrt{x}+4} dx = 2t^2 \frac{1-t}{t+4} 2t dt = 4t^3 \frac{1-t}{t+4} dt \] Does this yield the RHS? Not immediately apparent. But checking the bounds and the forms, they look **not equal**. **Answer:** **False** --- ### (c) \(\int_{1/2}^1 \frac{\sqrt{81x^8-9x^4+1}}{3x} dx = \int_1^2 \frac{\sqrt{81-t^8-9t^4+t^8}}{3t^5} dt\) Notice the integrands: - \( 81-t^8-9t^4+t^8 = 81 - 9t^4 \) - But \( 81x^8-9x^4+1 \) vs. \( 81 - 9t^4 \) after substitution is not matching. So, **False** --- ### (d) For any function \( f(x) \) defined on the whole real line, the parameterization \( x(t) = \ln t, y(t) = f(\ln t), t>0 \) gives the same curve as \( y = f(x) \). If \( x = \ln t \), then \( t = e^x \), so \( y = f(\ln t) = f(x) \). So the set of points is the same. **Answer:** **True** --- ### (e) The areas of the surfaces formed by \( y=f(x) \) and \( y=a-f(x) \) over \( 0 \leq x \leq b \) rotated about the y-axis are equal for any real \( a \) and every \( f(x) \). This is **not true** in general. The surfaces are generally different unless \( f(x) \) is symmetric in some way. **Answer:** **False** --- ### (f) The surface area of a solid with finite volume is always finite. Counterexample: Gabriel's Horn has finite volume but infinite surface area. **Answer:** **False** --- ### (g) \( r = a\cos\theta + b\sin\theta \) is the equation of an ellipse in polar coordinates. Yes, it's the general form of an ellipse with the center at the origin. **Answer:** **True** --- ### (h) The series \( \sum_{n=k}^{\infty} e^{-kn} \) converges for any fixed positive integer \( k \). For \( k>0 \), this is a geometric series with ratio \( e^{-k} < 1 \), so it **converges**. **Answer:** **True** --- ### (i) A sequence of Riemann sums of an increasing continuous function on \([a, b]\) (index variable \( n \) = number of subintervals \( \to \infty \)) is monotone. For increasing functions, left-hand sums increase, right-hand sums decrease as \( n \) increases, but the general sequence of Riemann sums (arbitrary sample points) **need not be monotone**. **Answer:** **False** --- ## **Final Answers:** **(a) False** **(b) False** **(c) False** **(d) True** **(e) False** **(f) False** **(g) True** **(h) True** **(i) False**