ForumTotal.com - Mathematics Help, Problems & Solutions

How can i prove sin(x)/x tends to 1 without circular unit circle reasoning?

Fri, 17 Apr 2026 14:47:26 +0000

I keep getting stuck when I try to prove that the limit of sin(x)/x as x approaches 0 is 1. I understand the squeeze theorem conceptually, but when I try to set up the inequalities using the unit circle and areas, my geometric argument for why sin(x) < x < tan(x) feels circular. I'm not sure if my justification for the area of a sector is actually relying on the limit I'm trying to prove.

How can I apply the Leibniz test if an alternating series doesn't decrease?

Fri, 17 Apr 2026 11:54:39 +0000

I’ve been working through some problems on the convergence of infinite series, and I keep getting stuck when a series has terms that alternate in sign but don’t strictly decrease in absolute value from the very first term. It makes applying the Leibniz test for alternating series feel a bit shaky to me.

How do I solve exponent equations like 3^(2x+1) = 7^(x-2) using logs?

Fri, 10 Apr 2026 15:10:37 +0000

I keep getting stuck when I try to solve for x in equations like 3^(2x+1) = 7^(x-2). I understand the basic log rules, but applying them to these more complicated exponents always throws me off my stride. I'm never quite sure if I should take the natural log of both sides right away or try to rewrite the bases first.

How do I set bounds for the washer method with y=x² and y=2x?

Fri, 10 Apr 2026 13:52:32 +0000

I’m working through a calculus problem where I need to find the volume of a solid of revolution, and I’ve set up my integral using the washer method. I keep getting a different answer than the textbook, and I think my error is in determining the correct limits of integration for the region bounded by y = x² and y = 2x. How do you properly establish those bounds when the curves intersect?

How can i avoid mistakes when choosing u and dv in integration by parts?

Fri, 10 Apr 2026 12:24:37 +0000

I'm working through some integration by parts problems and I keep getting stuck when the integrand is a product like x times a trigonometric function. I tried to solve ∫ x sin(x) dx and my final answer had the sign wrong on the cosine term. I think I'm making a mistake when I choose my u and dv, or maybe when I do the subsequent differentiation and integration.

Why is the induction step for the sum of the first n odd numbers unclear?

Fri, 10 Apr 2026 10:53:22 +0000

I keep getting stuck trying to prove that the sum of the first n odd numbers is n². I understand the pattern works, but my induction step feels like I'm just forcing it algebraically without really seeing why it holds together.

How do i show my decreasing sequence converges to its infimum?

Wed, 08 Apr 2026 21:51:52 +0000

I'm working through a problem about the convergence of a sequence defined by a recurrence relation, and I've hit a wall trying to apply the monotone convergence theorem. My sequence is decreasing and bounded below, so I know it should converge, but I'm stuck on formally proving the bound is the infimum.

What mistake am I making with shell vs washer method for y-axis rotation?

Wed, 08 Apr 2026 20:36:21 +0000

I’m working through a calculus problem where I need to find the volume of a solid formed by rotating a region around the y-axis. I set up my integral using the shell method, but when I check the answer key, they used the washer method instead. I can’t seem to get my shell method setup to match their final result, and I’m wondering if I’m misidentifying the radius or height of a cylindrical shell.

How do i solve x = 2 ln(x) for x?

Wed, 08 Apr 2026 19:01:26 +0000

I keep getting stuck when trying to solve for x in equations where it appears both inside and outside of a logarithm, like in x = 2 ln(x). I understand the properties individually, but combining them into a single algebraic solution has me completely stumped.

How do I pick the shell radius for this region rotated about the y-axis?

Wed, 08 Apr 2026 17:26:44 +0000

I'm working through a problem about finding the volume of a solid of revolution using cylindrical shells, and I'm stuck on setting up the integral. The region is bounded by y = sqrt(x), y=0, and x=4, rotated about the y-axis. I keep getting confused about whether the radius of a shell is just 'x' or if it's something else when the axis is vertical.

How do i set bounds for the washer method with y=x^2 and y=2x around the x-axis?

Wed, 08 Apr 2026 16:01:39 +0000

I’m working through a problem where I need to find the volume of a solid formed by rotating the region between y = x² and y = 2x around the x-axis. I set up my washer method integral with outer radius 2x and inner radius x², but when I evaluate it from 0 to 2, my answer doesn’t match the solution in the book. I think my understanding of the bounds might be off, or maybe I’m misidentifying the radii.

Why is my washer method volume negative, could it be the limits or squaring?

Mon, 06 Apr 2026 18:01:36 +0000

I'm working through a problem about finding the volume of a solid of revolution using the washer method, and I've set up my integral with the correct outer and inner radii. But when I evaluate it, I keep getting a negative result, which doesn't make sense for a volume. I must be messing up the limits of integration or the order of the functions when I square them.

Should I expect the same volume if pi is factored out in the disk method?

Mon, 06 Apr 2026 16:33:19 +0000

I'm working through a problem about finding the volume of a solid of revolution using the disk method, and I've set up my integral with the radius in terms of x. But when I evaluate it, my answer has a constant pi term, while the textbook solution has the constant factored out front. Is my approach wrong, or is this just a different way of writing the same final numerical value?

Why do shell and disk methods give different volumes for the same solid?

Mon, 06 Apr 2026 15:07:24 +0000

I'm working through a problem about finding the volume of a solid of revolution using the shell method, but I keep getting a different answer than when I use the disk method for the same region. I've checked my bounds and my integrals several times, and I'm starting to wonder if my understanding of the radius for the cylindrical shells is off.

How to set up disk method for region between y=x^2 and y=4 about the x-axis?

Mon, 06 Apr 2026 13:48:16 +0000

I'm working through a problem about finding the volume of a solid of revolution using the disk method, but I'm getting stuck on setting up the integral for the region bounded by y = x² and y = 4. My limits of integration seem off because when I solve for x I get ±2, but I'm not sure if that's correct for revolving around the x-axis.